Rainbow Connection in Hypergraphs - cantab.net rc hypergraphs slides.pdf · Rainbow Connection in...

Rainbow Connection in Hypergraphs

Henry Liu

Universidade Nova de Lisboa, Portugal

Joint work with Rui Carpentier, Manuel Silva and Teresa Sousa

Bordeaux Graph Workshop, 21 to 24 November 2012

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Introduction

All graphs and hypergraphs are simple and finite.

DefinitionThe rainbow connection number rc(G ) of a connected graph G isthe minimum number of colours needed to colour the edges of Gsuch that, any two vertices are connected by a path with distinctcolours (i.e., a rainbow path).

The function rc(G ) was first introduced by Chartrand, Johns,McKeon and Zhang in 2008.

The study of rc(G ) has since attracted a lot of interest. Manygeneralisations and variant functions have been considered.Recently, a survey by Li, Shi and Sun, and a book by Li and Sun,were published on the rainbow connection subject.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Introduction

All graphs and hypergraphs are simple and finite.

DefinitionThe rainbow connection number rc(G ) of a connected graph G isthe minimum number of colours needed to colour the edges of Gsuch that, any two vertices are connected by a path with distinctcolours (i.e., a rainbow path).

The function rc(G ) was first introduced by Chartrand, Johns,McKeon and Zhang in 2008.

The study of rc(G ) has since attracted a lot of interest. Manygeneralisations and variant functions have been considered.Recently, a survey by Li, Shi and Sun, and a book by Li and Sun,were published on the rainbow connection subject.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Introduction

All graphs and hypergraphs are simple and finite.

DefinitionThe rainbow connection number rc(G ) of a connected graph G isthe minimum number of colours needed to colour the edges of Gsuch that, any two vertices are connected by a path with distinctcolours (i.e., a rainbow path).

The function rc(G ) was first introduced by Chartrand, Johns,McKeon and Zhang in 2008.

The study of rc(G ) has since attracted a lot of interest. Manygeneralisations and variant functions have been considered.Recently, a survey by Li, Shi and Sun, and a book by Li and Sun,were published on the rainbow connection subject.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Introduction

All graphs and hypergraphs are simple and finite.

DefinitionThe rainbow connection number rc(G ) of a connected graph G isthe minimum number of colours needed to colour the edges of Gsuch that, any two vertices are connected by a path with distinctcolours (i.e., a rainbow path).

The function rc(G ) was first introduced by Chartrand, Johns,McKeon and Zhang in 2008.

The study of rc(G ) has since attracted a lot of interest. Manygeneralisations and variant functions have been considered.Recently, a survey by Li, Shi and Sun, and a book by Li and Sun,were published on the rainbow connection subject.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Example (Chartrand et al., 2008)

rc(G ) = e(G ) if and only if G is a tree.

Example (Chartrand et al., 2008)

rc(C3) = 1, and rc(Cn) = dn2e if n ≥ 4.

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C6

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C7

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Example (Chartrand et al., 2008)

rc(G ) = e(G ) if and only if G is a tree.

Example (Chartrand et al., 2008)

rc(C3) = 1, and rc(Cn) = dn2e if n ≥ 4.

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C6

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C7

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Example (Chartrand et al., 2008)

rc(G ) = e(G ) if and only if G is a tree.

Example (Chartrand et al., 2008)

rc(C3) = 1, and rc(Cn) = dn2e if n ≥ 4.

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C6

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C7

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.

But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

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v1 v2e1 . . . ...................................................................................................................................................................

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v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

v1 v2e1

. . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

v1 v2e1 . . . ...................................................................................................................................................................

..................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................

v3e2

. .............................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

v1 v2e1 . . . ...................................................................................................................................................................

..................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................

v3e2 . .............................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

.... ...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

v1 v2e1 . . . ...................................................................................................................................................................

..................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................

v3e2 . .............................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionFor 1 ≤ s < r , an (r , s)-path is an r-uniform interval hypergraphwhere every two consecutive edges intersect in s vertices.

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(5, 2)-path of length 3

DefinitionThe (r , s)-rainbow connection number rc(H, r , s) of a connectedhypergraph H is the minimum number of colours needed to colourthe edges of H such that, any two vertices are connected by arainbow (r , s)-path (if the minimum exists).

Hence, we have two generalisations of rc(G ) to hypergraphs.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionFor 1 ≤ s < r , an (r , s)-path is an r-uniform interval hypergraphwhere every two consecutive edges intersect in s vertices.

. . . . . . ........................................................................................................................................................................

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(5, 2)-path of length 3

DefinitionThe (r , s)-rainbow connection number rc(H, r , s) of a connectedhypergraph H is the minimum number of colours needed to colourthe edges of H such that, any two vertices are connected by arainbow (r , s)-path (if the minimum exists).

Hence, we have two generalisations of rc(G ) to hypergraphs.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionFor 1 ≤ s < r , an (r , s)-path is an r-uniform interval hypergraphwhere every two consecutive edges intersect in s vertices.

. . . . . . ........................................................................................................................................................................

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(5, 2)-path of length 3

DefinitionThe (r , s)-rainbow connection number rc(H, r , s) of a connectedhypergraph H is the minimum number of colours needed to colourthe edges of H such that, any two vertices are connected by arainbow (r , s)-path (if the minimum exists).

Hence, we have two generalisations of rc(G ) to hypergraphs.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionFor 1 ≤ s < r , an (r , s)-path is an r-uniform interval hypergraphwhere every two consecutive edges intersect in s vertices.

. . . . . . ........................................................................................................................................................................

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(5, 2)-path of length 3

DefinitionThe (r , s)-rainbow connection number rc(H, r , s) of a connectedhypergraph H is the minimum number of colours needed to colourthe edges of H such that, any two vertices are connected by arainbow (r , s)-path (if the minimum exists).

Hence, we have two generalisations of rc(G ) to hypergraphs.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

Remark

It is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

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rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

Remark

It is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

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.............................................................................................................................................

.....................................................................................................................................................................

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.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

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.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

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.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

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.....................................................................................................................................................................

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.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2

e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

. . . ..............................................................................................................................................................................................................

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.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

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.....................................................................................................................................................................................................................................................................................

....................................................................................

................................................................................................

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

.....................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................

....................................................................................

................................................................................................

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

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.....................................................................................................................................................................................................................................................................................

....................................................................................

................................................................................................

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

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rc(Crn, r , s) well-defined for all 1 ≤ s < r . We consider r ≥ 3.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

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rc(Crn, r , s) well-defined for all 1 ≤ s < r . We consider r ≥ 3.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 2 (CLSS, 2012)

Let n > r ≥ 3. Then for sufficiently large n,

(a) rc(Crn) = rc(Cr

n, r , 1) = d n2(r−1)e.

(b) rc(Crn, r , r − 1) = dn2e.

(c) rc(Crn, r , s) ∈ d , d + 1 for 1 ≤ s ≤ r − 2, where

d = dn+1−2s2(r−s) e, the “ (r , s)-diameter of Cr

n”.

Hence, rc(Crn, r , s) = d n

2(r−s)e+ O(1) for all 1 ≤ s < r .

Proof (sketch).

Lower bound: Computation of d is easy ⇒ (a) (except for n ≡ 1(mod 2(r − 1))) and (c). Remaining lower bounds also easy.

Upper bound: Colour a “wraparound” (r , s)-path with the requirednumber of colours.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 2 (CLSS, 2012)

Let n > r ≥ 3. Then for sufficiently large n,

(a) rc(Crn) = rc(Cr

n, r , 1) = d n2(r−1)e.

(b) rc(Crn, r , r − 1) = dn2e.

(c) rc(Crn, r , s) ∈ d , d + 1 for 1 ≤ s ≤ r − 2, where

d = dn+1−2s2(r−s) e, the “ (r , s)-diameter of Cr

n”.

Hence, rc(Crn, r , s) = d n

2(r−s)e+ O(1) for all 1 ≤ s < r .

Proof (sketch).

Lower bound: Computation of d is easy ⇒ (a) (except for n ≡ 1(mod 2(r − 1))) and (c). Remaining lower bounds also easy.

Upper bound: Colour a “wraparound” (r , s)-path with the requirednumber of colours.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 2 (CLSS, 2012)

Let n > r ≥ 3. Then for sufficiently large n,

(a) rc(Crn) = rc(Cr

n, r , 1) = d n2(r−1)e.

(b) rc(Crn, r , r − 1) = dn2e.

(c) rc(Crn, r , s) ∈ d , d + 1 for 1 ≤ s ≤ r − 2, where

d = dn+1−2s2(r−s) e, the “ (r , s)-diameter of Cr

n”.

Hence, rc(Crn, r , s) = d n

2(r−s)e+ O(1) for all 1 ≤ s < r .

Proof (sketch).

Lower bound: Computation of d is easy ⇒ (a) (except for n ≡ 1(mod 2(r − 1))) and (c). Remaining lower bounds also easy.Upper bound: Colour a “wraparound” (r , s)-path with the requirednumber of colours.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Complete Multipartite Hypergraphs

DefinitionFor t ≥ r ≥ 2 and 1 ≤ n1 ≤ · · · ≤ nt , the r -uniform hypergraphKr

n1,...,ntis ...

.. ..

. ......

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K32,2,3,4

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Krn1,...,nt

is a complete multipartite hypergraph.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Complete Multipartite Hypergraphs

DefinitionFor t ≥ r ≥ 2 and 1 ≤ n1 ≤ · · · ≤ nt , the r -uniform hypergraphKr

n1,...,ntis ...

.. ..

. ......

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K32,2,3,4

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.................................................................................... ...........................................................................................................................

.......

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...............................................................

..........................

..................................................................

Krn1,...,nt

is a complete multipartite hypergraph.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Complete Multipartite Hypergraphs

DefinitionFor t ≥ r ≥ 2 and 1 ≤ n1 ≤ · · · ≤ nt , the r -uniform hypergraphKr

n1,...,ntis ...

.. ..

. ......

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..............

............................................

..........................................................

..............

............................................

..........................................................

..............

............................................

..........................................................

..............

............................................

..........................................................

.........................................................................................................

..............................................................................................................................................................

.............................................................................................................................................................

K32,2,3,4

.................................................................................................................................................................................................

.................................................................................... ...........................................................................................................................

.......

.................................

...............................................................

..........................

..................................................................

Krn1,...,nt

is a complete multipartite hypergraph.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Complete Multipartite Hypergraphs

DefinitionFor t ≥ r ≥ 2 and 1 ≤ n1 ≤ · · · ≤ nt , the r -uniform hypergraphKr

n1,...,ntis ...

.. ..

. ......

.................................................................................

..........................................................................................................................

.........................................................................................................................

..............

............................................

..........................................................

..............

............................................

..........................................................

..............

............................................

..........................................................

..............

............................................

..........................................................

.........................................................................................................

..............................................................................................................................................................

.............................................................................................................................................................

K32,2,3,4

.................................................................................................................................................................................................

.................................................................................... ...........................................................................................................................

.......

.................................

...............................................................

..........................

..................................................................

Krn1,...,nt

is a complete multipartite hypergraph.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

rc(K2n1,...,nt

) determined by Chartrand et al. Let r ≥ 3.

Theorem 3 (CLSS, 2012)

Let t ≥ r ≥ 3 and 1 ≤ n1 ≤ · · · ≤ nt . Then,

rc(Krn1,...,nt

) =

1 if nt = 1,

2 if nt−1 ≥ 2, or t > r , nt−1 = 1 and nt ≥ 2,

nt if t = r and nt−1 = 1.

Proof not too difficult. For the last case, Krn1,...,nt

is minimallyconnected.More interesting to consider rc(Kr

n1,...,nt, r , s).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

rc(K2n1,...,nt

) determined by Chartrand et al. Let r ≥ 3.

Theorem 3 (CLSS, 2012)

Let t ≥ r ≥ 3 and 1 ≤ n1 ≤ · · · ≤ nt . Then,

rc(Krn1,...,nt

) =

1 if nt = 1,

2 if nt−1 ≥ 2, or t > r , nt−1 = 1 and nt ≥ 2,

nt if t = r and nt−1 = 1.

Proof not too difficult. For the last case, Krn1,...,nt

is minimallyconnected.More interesting to consider rc(Kr

n1,...,nt, r , s).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

rc(K2n1,...,nt

) determined by Chartrand et al. Let r ≥ 3.

Theorem 3 (CLSS, 2012)

Let t ≥ r ≥ 3 and 1 ≤ n1 ≤ · · · ≤ nt . Then,

rc(Krn1,...,nt

) =

1 if nt = 1,

2 if nt−1 ≥ 2, or t > r , nt−1 = 1 and nt ≥ 2,

nt if t = r and nt−1 = 1.

Proof not too difficult. For the last case, Krn1,...,nt

is minimallyconnected.

More interesting to consider rc(Krn1,...,nt

, r , s).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

rc(K2n1,...,nt

) determined by Chartrand et al. Let r ≥ 3.

Theorem 3 (CLSS, 2012)

Let t ≥ r ≥ 3 and 1 ≤ n1 ≤ · · · ≤ nt . Then,

rc(Krn1,...,nt

) =

1 if nt = 1,

2 if nt−1 ≥ 2, or t > r , nt−1 = 1 and nt ≥ 2,

nt if t = r and nt−1 = 1.

Proof not too difficult. For the last case, Krn1,...,nt

is minimallyconnected.More interesting to consider rc(Kr

n1,...,nt, r , s).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 4 (CLSS, 2012)

Let t ≥ r ≥ 3, s ≤ r − 2 and 1 ≤ n1 ≤ · · · ≤ nt be such thatnt = 1 or n2(t−r)+s+1 ≥ 2. Then,

rc(Krn1,...,nt

, r , s) =

1 if nt = 1,

2 if nt ≥ 2.

Proof still not difficult. But surprisingly, rc(Krn1,...,nt

, r , r − 1) is alot more difficult to determine ...

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 4 (CLSS, 2012)

Let t ≥ r ≥ 3, s ≤ r − 2 and 1 ≤ n1 ≤ · · · ≤ nt be such thatnt = 1 or n2(t−r)+s+1 ≥ 2. Then,

rc(Krn1,...,nt

, r , s) =

1 if nt = 1,

2 if nt ≥ 2.

Proof still not difficult. But surprisingly, rc(Krn1,...,nt

, r , r − 1) is alot more difficult to determine ...

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Theorem 5 (CLSS, 2012)

Let t ≥ r ≥ 3, 1 ≤ n1 ≤ · · · ≤ nt , n = nt and b =∑

S

∏i∈S ni ,

where S ranges over all subsets of 1, . . . , t − 1 with size r − 1.Then,

rc(Krn1,...,nt

, r , r − 1) =

1 if nt = 1,

d b√

n e if t = r and n1 = 1,

min(d b√

n e, r + 2) if t = r and n1 ≥ 2,

min(d b√

n e, 3) if t > r .

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.

rc(Krn1,...,nt

, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; andrc(Kr

n1,...,nt, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c

. . .

.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3).

Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples

⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.

Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

............................................................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

.............................................................................................................................................

.....................................................................................................................................................................

......................................................................................................

......................................................................................................

.................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................

n1 n2 · · · nt−1

nt

.........................................................................................................................................................................................................

.........................................................................................................................................................................................................

............................................................................................................................................................................................

.........................................................................................................................................................................................................

.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Separation Results

Theorem 6 (CLSS, 2012)

Let a > 0, r ≥ 3 and 1 ≤ s 6= s ′ < r . Then, there exists anr-uniform hypergraph H such that rc(H, r , s)− rc(H) ≥ a, andthere exists an r-uniform hypergraph H such thatrc(H, r , s)− rc(H, r , s ′) ≥ a.

Proof.First part with s > 1, and second part with s > s ′, take H = Cr

n.⇒ Difference is at least

n

2(r − s)− n

2(r − s ′)+ O(1) ≥ Ω(n).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Separation Results

Theorem 6 (CLSS, 2012)

Let a > 0, r ≥ 3 and 1 ≤ s 6= s ′ < r . Then, there exists anr-uniform hypergraph H such that rc(H, r , s)− rc(H) ≥ a, andthere exists an r-uniform hypergraph H such thatrc(H, r , s)− rc(H, r , s ′) ≥ a.

Proof.First part with s > 1, and second part with s > s ′, take H = Cr

n.⇒ Difference is at least

n

2(r − s)− n

2(r − s ′)+ O(1) ≥ Ω(n).

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

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......................................................................................................................................................................................

...................................................................................................................................................................................................................

.............................................. .......

....................................................................................................................................................................... .......

. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

...............................

.......................................

.......................................

...............................

...............................

....................................................................................................................................................................... ....................................................................................................................................................

.................................................................... ................................................................................................. ...........................................

................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................

Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

...................................................................................................................................

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...................................................................................................................................................................................................................

.............................................. .......

....................................................................................................................................................................... .......

. .s

r

u v

...................................................................................... ....... .................................................................. .......s ′

r − s ′ − 1,“fixed”

...............................

.......................................

.......................................

...............................

...............................

....................................................................................................................................................................... ....................................................................................................................................................

.................................................................... ................................................................................................. ...........................................

................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................

Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

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...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

.............................................. .......

....................................................................................................................................................................... .......

. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

...............................

.......................................

.......................................

...............................

...............................

....................................................................................................................................................................... ....................................................................................................................................................

.................................................................... ................................................................................................. ...........................................

................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................

Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

...................................................................................................................................

......................................................................................................................................................................................

...................................................................................................................................................................................................................

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......................................................................................................................................................................................

...................................................................................................................................................................................................................

.............................................. .......

....................................................................................................................................................................... .......

. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

...............................

.......................................

.......................................

...............................

...............................

....................................................................................................................................................................... ....................................................................................................................................................

.................................................................... ................................................................................................. ...........................................

................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................

Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.

Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

...................................................................................................................................

......................................................................................................................................................................................

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......................................................................................................................................................................................

...................................................................................................................................................................................................................

.............................................. .......

....................................................................................................................................................................... .......

. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

...............................

.......................................

.......................................

...............................

...............................

....................................................................................................................................................................... ....................................................................................................................................................

.................................................................... ................................................................................................. ...........................................

................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................

Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

References

1. G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, Rainbowconnection in graphs, Math. Bohem. 133(1) (2008), 85-98.

2. X. Li, Y. Shi, and Y. Sun, Rainbow connections of graphs - asurvey, Graphs Combin., to appear.

3. X. Li, and Y. Sun, “Rainbow Connections of Graphs”,Springer-Verlag, New York, 2012, viii+103pp.

Henry Liu Rainbow Connection in Hypergraphs

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Thank you!

Henry Liu Rainbow Connection in Hypergraphs