Bounded Degree Subgraphs of Dense Graphs

Bounded Degree Subgraphs of Dense Graphs

Julia Bottcher

Technische Universitat Munchen

13th International Conference on Random Structures and Algorithms,May 28-June 1, 2007, Tel Aviv

(joint work with Mathias Schacht & Anusch Taraz)

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Subgraph containment problem

Question

Given a graph H, which conditions on an n-vertex graph G = (V , E)

ensure H ⊆ G?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Subgraph containment problem

Question

Given a graph H, which conditions on an n-vertex graph G = (V , E)

ensure H ⊆ G?

Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Subgraph containment problem

Question

Given a graph H, which conditions on an n-vertex graph G = (V , E)

ensure H ⊆ G?

Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.

H of small/fixed size

Erdos–Stone: δ(G) ≥(

χ(H)−2χ(H)−1 + o(1)

)

n =⇒ H ⊆ G.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Subgraph containment problem

Question

Given a graph H, which conditions on an n-vertex graph G = (V , E)

ensure H ⊆ G?

Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.

H of small/fixed size

Erdos–Stone: δ(G) ≥(

χ(H)−2χ(H)−1 + o(1)

)

n =⇒ H ⊆ G.

This talk

H is a spanning subgraph of G.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Subgraph containment problem

Question

Given a graph H, which conditions on an n-vertex graph G = (V , E)

ensure H ⊆ G?

Our aim: every graph G = (V , E) with minimum degree δ(G) ≥??contains a given graph H.

Classical example:

δ(G) ≥ 12n ⇒ Ham ⊆ G DIRAC’52

This talk

H is a spanning subgraph of G.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

From Small Graphs to Spanning Graphs

A graph H of constant size is forced in G, when

δ(G) ≥

(

χ(H) − 2χ(H) − 1

+ o(1)

)

n.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

From Small Graphs to Spanning Graphs

A graph H of constant size is forced in G, when

δ(G) ≥

(

χ(H) − 2χ(H) − 1

+ o(1)

)

n.

For spanning H we need at least δ(G) ≥ χ(H)−1χ(H)

n, because:

H6⊆ G

n3 − 1

n3 − 1

n3 + 2

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

From Small Graphs to Spanning Graphs

A graph H of constant size is forced in G, when

δ(G) ≥

(

χ(H) − 2χ(H) − 1

+ o(1)

)

n.

For spanning H we need at least δ(G) ≥ χ(H)−1χ(H)

n, because:

H6⊆ G

n3 − 1

n3 − 1

n3 + 2

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ r−1r n ⇒ ⌊n

r ⌋ disj. copies of Kr ⊆ G.HAJNAL,SZEMEREDI’69

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ 23n ⇒

replacements⊆ G

HAJNAL,SZEMEREDI’69

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ 23n ⇒

replacements⊆ G

HAJNAL,SZEMEREDI’69

δ(G) ≥ χ(F)−1χ(F)

n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G

Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ 23n ⇒

replacements⊆ G

HAJNAL,SZEMEREDI’69

δ(G) ≥ χ(F)−1χ(F)

n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G

Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01

δ(G) ≥ r−1r n ⇒ (Ham)r ⊆ G.

FAN, KIERSTEAD ’95

KOMLOS, SARKOZY, AND SZEMEREDI’98

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ 23n ⇒

replacements⊆ G

HAJNAL,SZEMEREDI’69

δ(G) ≥ χ(F)−1χ(F)

n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G

Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01

δ(G) ≥ 23n ⇒ Ham2 ⊆ G

Posa’s conjecture FAN, KIERSTEAD ’95

KOMLOS, SARKOZY, AND SZEMEREDI’98

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Big Graphs

Does δ(G) ≥(

χ(H)−1χ(H)

+ o(1))

n imply H ⊆ G?

δ(G) ≥ 23n ⇒

replacements⊆ G

HAJNAL,SZEMEREDI’69

δ(G) ≥ χ(F)−1χ(F)

n ⇒ ⌊ n|F |⌋ disj. copies of F ⊆ G

Alon-Yuster conjecture KOMLOS, SARKOZY, AND SZEMEREDI ’01

δ(G) ≥ 23n ⇒ Ham2 ⊆ G

Posa’s conjecture FAN, KIERSTEAD ’95

KOMLOS, SARKOZY, AND SZEMEREDI’98

δ(G) ≥ (12 + γ)n ⇒ every spanning tree with ∆(T ) ≤ cn

log n ⊆ G.

Tree universality KOMLOS, SARKOZY, AND SZEMEREDI’95

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Generalizing Conjecture

Naıve conjecture

HG

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Generalizing Conjecture

Naıve conjecture

H

χ(H) = k

G

δ(G) ≥ ( k−1k + γ)n

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Generalizing Conjecture

Naıve conjecture

H

χ(H) = k

∆(H) ≤ ∆

G

δ(G) ≥ ( k−1k + γ)n

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Generalizing Conjecture

Naıve conjecture

For all k , ∆ ≥ 1, and γ > 0 exists n0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Generalizing Conjecture

Naıve conjecture

For all k , ∆ ≥ 1, and γ > 0 exists n0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Counterexample:

H : random bipartite graph on n2 + n

2 vertices with ∆(H) ≤ ∆.

G : two cliques of size(

12 + γ

)

n sharing 2γn vertices.

K(12 +γ)nK(1

2 +γ)n

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Conjecture of Bollobas and Komlos

Conjecture

For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Conjecture of Bollobas and Komlos

Theorem

For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

bw(H) ≤ βn

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Bandwidth:

bw(G) ≤ b if there is a labelling of V (G) by 1, . . . , ns.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.

i j

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Conjecture of Bollobas and Komlos

Theorem

For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

bw(H) ≤ βn

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Examples for H:

Hamiltonian cycles (bandwidth 2)

graphs of constant tree width (bandwidth O(n/ log∆ n))

bounded degree planar graphs (bandwidth O(n/ log∆ n))

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Conjecture of Bollobas and Komlos

Theorem

For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t.

H

χ(H) = k

∆(H) ≤ ∆

bw(H) ≤ βn

G

δ(G) ≥ ( k−1k + γ)n

=⇒ G contains H.

Abbasi ’98 annouced 2-chromatic case (see also Hie. p Han ’06)

additional γn is necessary

Proof uses regularity lemma, blow-up lemma, and affirmative solutionof Posa’s conjecture

G ⊇

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Conjecture of Bollobas and Komlos

χ(H) =k , ∆(H) ≤ ∆ ,bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Naıve conjecture

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (2−12 + γ)n =⇒ G contains H

Counterexample:

H : random bipartite graph on n2 + n

2 vertices with ∆(H) ≤ ∆.

G : two cliques of size(

12 + γ

)

n sharing 2γn vertices.

K(12 +γ)nK(1

2 +γ)n

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Increasing the minimum degree

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (3−13 + γ)n =⇒ G contains H

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Increasing the minimum degree

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (3−13 + γ)n =⇒ G contains H

Counterexample:

H ′ : random bipartite graph on n4 + n

4 vertices with ∆(H) ≤ ∆3 .

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Increasing the minimum degree

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (3−13 + γ)n =⇒ G contains H

Counterexample:

H ′ : random bipartite graph on n4 + n

4 vertices with ∆(H) ≤ ∆3 .

H

H ′H ′H ′

n4

n4

n4

n4

G

n3

n3

n3

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Increasing the minimum degree

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (3−13 + γ)n =⇒ G contains H

Counterexample:

H ′ : random bipartite graph on n4 + n

4 vertices with ∆(H) ≤ ∆3 .

H

H ′H ′H ′

n4

n4

n4

n4

G

n3

n3

n3

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Increasing the minimum degree

χ(H) =2, ∆(H) ≤ ∆

δ(G) ≥ (3−13 + γ)n =⇒ G contains H

Counterexample:

H ′ : random bipartite graph on n4 + n

4 vertices with ∆(H) ≤ ∆3 .

H

H ′H ′H ′

n4

n4

n4

n4

G

n3

n3

n3

Obstacle: G has big subsets A and B with e(A, B) empty.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

Because:

H has an ∆ + 1-colouring with colour classes of equal size. (THM OF H.SZ.)

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

Because:

H has an ∆ + 1-colouring with colour classes of equal size. (THM OF H.SZ.)

Every equi-partition of G into ∆ + 1 parts is ε-regular.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

Because:

H has an ∆ + 1-colouring with colour classes of equal size. (THM OF H.SZ.)

Every equi-partition of G into ∆ + 1 parts is ε-regular.

There is such a partition that is also super-regular.

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

Because:

H has an ∆ + 1-colouring with colour classes of equal size. (THM OF H.SZ.)

Every equi-partition of G into ∆ + 1 parts is ε-regular.

There is such a partition that is also super-regular.

By blow-up lemma, G contains H. (KOMLOS, SARKOZY, SZEMEREDI)

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

What about sparser graphs?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Quasi-random graphs

Obstacle: G has big subsets A and B with e(A, B) empty.

(n, d , λ)-graphs G (i.e. d-regular with λ(G) ≤ λ) have∣

∣e(A, B) −dn

|A||B|∣

∣ ≤ λ√

|A||B| for all A, B ⊆ V (G).

Observation

For d = γn and λ = o(n) every (sufficiently large) (n, d , λ)-graph Gcontains all H on n vertices with ∆(H) ≤ ∆.

What about sparser graphs?

If λ(G) = o(

d3

n2 log n

)

(i.e. d ≫ n4/5 log2/5 n) then G has a triangle

factor. KRIVELEVICH, SUDAKOV, SZABO’04

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Which further restrictions on G allow us to ommit the bandwidthrestriction on H?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Which further restrictions on G allow us to ommit the bandwidthrestriction on H?

How many copies of H do we get?(for bipartite H see Person’07)

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Which further restrictions on G allow us to ommit the bandwidthrestriction on H?

How many copies of H do we get?(for bipartite H see Person’07)

What about spanning graphs H of non-constant max. degree?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Which further restrictions on G allow us to ommit the bandwidthrestriction on H?

How many copies of H do we get?(for bipartite H see Person’07)

What about spanning graphs H of non-constant max. degree?

Julia Bottcher TU Munchen Tel Aviv ’07

Bounded Degree Subgraphs of Dense Graphs

Concluding remarks

χ(H) = k , ∆(H) ≤ ∆, bw(H) ≤ βn

δ(G) ≥ ( k−1k + γ)n =⇒ G contains H

Which further restrictions on G allow us to ommit the bandwidthrestriction on H?

How many copies of H do we get?(for bipartite H see Person’07)

What about spanning graphs H of non-constant max. degree?

Ba Łakashah!

Julia Bottcher TU Munchen Tel Aviv ’07