Guillem Perarnau SIAM Conference on Discrete Math ...web.mat.bham.ac.uk/G.Perarnau/slides/siam.pdf · Guillem Perarnau SIAM Conference on Discrete Math,Minneapolis- June 19th, 2014

Spanning F-free subgraphs with large minimum degree

Guillem Perarnau

SIAM Conference on Discrete Math, Minneapolis - June 19th, 2014

McGill University, Montreal, Canada

joint work with Bruce Reed.

The problem

Let G be a large graph and F a fixed one.

Does G contain a “large” F -free subgraph?

Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

The problem

Let G be a d-regular graph (d large enough) and F a fixed graph.

Does G contain a spanning F -free subgraph with large minimum degree?

Question


The problem



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The problem



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The problem



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The problem



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The problem



Question

f (d ,F ) := maxt : for every d-reg graph G there exists a spanningF -free subgraph with minimum degree ≥ t


The conjecture

Turan numbers: ex(n,F ) = maxe(G) : G subgraph of Kn and G is F -free

Since Kd+1 is d-regular we have,

f (d ,F ) ≤ 2ex(d + 1,F )

d + 1= O

(ex(d ,F )

d

).

For every fixed graph F ,

f (d ,F ) = Θ

(ex(d ,F )

d

).

Conjecture (Foucaud, Krivelevich, P. (2013))


Easy case: F = C3

Consider a partition V (G) = V1 ∪ V2 that maximizes e(V1,V2).

Let H be the bipartite subgraphcontaining the edges E(V1,V2).Then,

minimum degree of H is at leastd/2, and

H is C3-free.

f (d ,C3) =

(1

2+ o(1)

)d .


General case: F with χ(F ) = k

Consider a partition V (G) = V1 ∪ · · · ∪ Vk−1 that maximizes∑

i 6=j e(Vi ,Vj).

Let H be the (k − 1)-partite subgraphcontaining the edges ∪i 6=jE(Vi ,Vj).Then,

minimum degree of H is at least(1− 1

k−1

)d , and

H is F -free.

f (d ,F ) =

(1− 1

k − 1+ o(1)

)d .

That solves completely the case when χ(F ) ≥ 3.

If χ(F ) = 2 (F bipartite),f (d ,F ) = o(d) .


General case: F with χ(F ) = k

Consider a partition V (G) = V1 ∪ · · · ∪ Vk−1 that maximizes∑

i 6=j e(Vi ,Vj).

Let H be the (k − 1)-partite subgraphcontaining the edges ∪i 6=jE(Vi ,Vj).Then,

minimum degree of H is at least(1− 1

k−1

)d , and

H is F -free.

f (d ,F ) =

(1− 1

k − 1+ o(1)

)d .

That solves completely the case when χ(F ) ≥ 3.

If χ(F ) = 2 (F bipartite),f (d ,F ) = o(d) .


Bipartite case: F with χ(F ) = 2

Simplest case: F = T is a tree.Then ex(d ,T ) = O(d) and the conjecture states

ex(d ,T ) = Θ(1) .

Simplest non-trivial case: F = C4 is a cycle of length 4.Then ex(d ,C4) = O(d3/2) and we have the upper bound

ex(d ,C4) = O(√d) .

If d is large,

f (d ,C4) = Ω(d1/3

).

Theorem (Kun (2013))

If d is large,

f (d ,C4) = Ω

( √d

log d

).

Theorem (Foucaud, Krivelevich, P. (2013))


Bipartite case: F with χ(F ) = 2

Simplest case: F = T is a tree.Then ex(d ,T ) = O(d) and the conjecture states

ex(d ,T ) = Θ(1) .

Simplest non-trivial case: F = C4 is a cycle of length 4.Then ex(d ,C4) = O(d3/2) and we have the upper bound

ex(d ,C4) = O(√d) .

If d is large,

f (d ,C4) = Ω(d1/3

).

Theorem (Kun (2013))

If d is large,

f (d ,C4) = Ω

( √d

log d

).

Theorem (Foucaud, Krivelevich, P. (2013))


The theorem

If d is large,

f (d ,C4) = Θ(√

d).

Theorem (P., Reed (2014))

Is the regularity condition needed? Essentially yes.

Let G = Kδ,∆ (max degree ∆ \ min degree δ). Any C4-free subgraph H ⊆ Kδ,∆

has minimum degree δH = O(

δ√∆

)/. If δ ∆, then δ(H) = o(

√δ).


The theorem

If d is large,

f (d ,C4) = Θ(√

d).


Is the regularity condition needed? Essentially yes.

Let G = Kδ,∆ (max degree ∆ \ min degree δ). Any C4-free subgraph H ⊆ Kδ,∆

has minimum degree δH = O(

δ√∆

)/. If δ ∆, then δ(H) = o(

√δ).


Drawing the proof

Bipartize graph G : still minimum degree ≥ d/2.


Drawing the proof

Randomly color A


Drawing the proof

For a vertex a ∈ A keep color if . . .


Drawing the proof

For a vertex a ∈ A keep color if . . .. . . not many bad neighbors.


Drawing the proof

Then also remove dangerous edges.


Drawing the proof

For a vertex a ∈ A uncolor if . . .


Drawing the proof

For a vertex a ∈ A uncolor if . . .. . . too many bad neighbors.


Drawing the proof

Then, uncolor and keep all the edges


Drawing the proof

After first iteration we get a partial coloring.


Drawing the proof

We keep iterating . . .


Drawing the proof

We keep iterating . . .. . . until a small number of vertices are uncolored.


Drawing the proof

Then, color all them at once.


Drawing the proof

Important Properties: 1.- for every v ∈ B, N(v) is rainbow.2.- the minimum degree is Ω(d).


Drawing the proof

Color B in the same way.


Drawing the proof

Consider a extremal graph G without 4-cycles (V (G) = colors).Use the coloring on G to embed it onto G


Drawing the proof

Keep just the edges of G that agree with edges in G.


Drawing the proof

Fact I: because of the embedding, no rainbow 4-cycles.


Drawing the proof

Fact I: because of the embedding, no rainbow 4-cyclesFact II: because of the properties of the coloring, no non-rainbow 4-cycles.


Drawing the proof

The subgraph obtained is C4-free and has large minimum degree.


The real theorem

Let F = F1, . . . ,Fs be a family of fixed graphs.

We say that F is closed if for every F ∈ F and G

G is F -free ⇐⇒ no locally injective homomorphism from F to G .

Let F be a closed family and d large,

f (d ,F) = Θ

(ex(d ,F)

d

).


Examples:

cycles: F = C3, . . . ,C2r+1(existence of subgraphs with large girth andlarge minimum degree), F = C2p : p prime.complete bipartite graphs: for any ai , bi , i ≤ n, F = ∪iKai ,bi .

First unknown case: C8

THANKS FOR YOUR ATTENTION


The real theorem

Let F = F1, . . . ,Fs be a family of fixed graphs.

We say that F is closed if for every F ∈ F and G

G is F -free ⇐⇒ no locally injective homomorphism from F to G .

Let F be a closed family and d large,

f (d ,F) = Θ

(ex(d ,F)

d

).


Examples:

cycles: F = C3, . . . ,C2r+1(existence of subgraphs with large girth andlarge minimum degree), F = C2p : p prime.complete bipartite graphs: for any ai , bi , i ≤ n, F = ∪iKai ,bi .

First unknown case: C8

THANKS FOR YOUR ATTENTIONGuillem Perarnau Spanning F-free subgraphs with large minimum degree 9 / 9

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Guillem Perarnau SIAM Conference on Discrete Math ...web.mat.bham.ac.uk/G.Perarnau/slides/siam.pdf · Guillem Perarnau SIAM Conference on Discrete Math,Minneapolis- June 19th, 2014