Friday,July3,2015GOALSeminar-UniversitéLyon1 AurélieLagoutte Coloringgraphswithnoevenholeoflengthatleast6:thetriangle-freecase

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Motivation Result & Proof Conclusion

Coloring graphs with no even hole of length at least 6: the triangle-free case

Aurélie Lagoutte

LIP, ENS Lyon

Friday, July 3, 2015

GOAL Seminar - Université Lyon 1

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Proper coloring: two adjacent vertices get different colors.

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ω(G): size of the largest clique

χ(G): min. number of colors in a proper coloring

⇒χ(G)≥ω(G)

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⇒χ(G)≥ω(G)

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⇒χ(G)≥ω(G)

χ(C5)> ω(C5)!

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χ-boundedness

LetC be a hereditary class of graphs.

Definition (Gyárfás 1987)

The classC is χ-bounded if there exists f such that for every G∈ C,χ(G)≤f(ω(G)).

Perfect graphs are χ-bounded withf(x) =x. Triangle-free graphs is not a χ-bounded class.

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χ-boundedness

LetC be a hereditary class of graphs.

Definition (Gyárfás 1987)

The classC is χ-bounded if there exists f such that for every G∈ C,χ(G)≤f(ω(G)).

Perfect graphs are χ-bounded withf(x) =x.

Triangle-free graphs is not aχ-bounded class.

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Theorem (Erdős 1959)

For everyk, `, there exist graphs with girth≥k and chromatic number≥`.

⇒No hope to get a χ-bounded class by forbidding only finitely many cycles.

⇒What about forbidding a tree H? Conjecture (Gyárfás 1975)

The class ofH-free graphs isχ-bounded ifH is a tree. Proved when:

H is a path (Gyárfás 1987) H is a star

H has radius two (or three, with extra conditions)

H is any tree but ’H-free’ means no subdivision of H instead of no induced subgraphs isom. to H (Scott 1997).

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⇒What about forbidding a tree H? Conjecture (Gyárfás 1975)

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⇒What about forbidding a tree H?

Conjecture (Gyárfás 1975)

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The class ofH-free graphs isχ-bounded ifH is a tree.

Proved when:

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The class ofH-free graphs isχ-bounded ifH is a tree.

Proved when:

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Hole Parity & Length

Conjectures (Gyárfás 1987)

The class of graphs with no oddhole is χ-bounded.

For every k, the class of graphs with nolong hole is χ-bounded. (long = of length ≥k)

For every k, the class of graphs with nolong odd hole is χ-bounded. (long = of length ≥k)

First and second conjectures were proved.

(Scott, Seymour 2014 & Chudnovsky, Scott, Seymour, 2015) Triangle-free case of the third conjecture has just been proved (Scott, Seymour 2015)

: For everyk, there exists` such that every triangle-free graphG withχ(G)≥` has a sequence of holes ofk consecutive lengths.

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Hole Parity & Length

Conjectures (Gyárfás 1987)

The class of graphs with no oddhole is χ-bounded.

For every k, the class of graphs with nolong hole is χ-bounded. (long = of length ≥k)

For every k, the class of graphs with nolong odd hole is χ-bounded. (long = of length ≥k)

First and second conjectures were proved.

(Scott, Seymour 2014 & Chudnovsky, Scott, Seymour, 2015) Triangle-free case of the third conjecture has just been proved (Scott, Seymour 2015): For everyk, there exists` such that every triangle-free graphG withχ(G)≥` has a sequence of holes ofk consecutive lengths.

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Even-hole-free graphs

Well-understood class of graphs:

Decomposition theorem, recognition algorithm.

Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour 2008) Every even-hole-free graph has a bisimplicial vertex.

x is bisimplicial if N(x) is the union of two cliques.

⇒For every even-hole-free graph G,χ(G)≤2ω(G)−1.

v

N(v) G\N[v]

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v

N(v) G\N[v]

≤ω(G)−1

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Even-hole-free graphs

v

N(v) G\N[v]

≤ω(G)−1

≤ω(G)−1 d(v)≤2ω(G)−2

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Even-hole-free graphs

v

N(v) G\N[v]

≤ω(G)−1

≤ω(G)−1 d(v)≤2ω(G)−2

By ind.χ(G⁰)≤2ω(G)−1

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Even-hole-free graphs

v

N(v) G\N[v]

≤ω(G)−1

≤ω(G)−1 d(v)≤2ω(G)−2

By ind.χ(G⁰)≤2ω(G)−1 One color available

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Forbidding C

₄

?

Letk be an integer. Consider the classC_k of graphs with:

No triangle No induced C₄

No induced cycle of length divisible byk

In particular: noC4 subgraphs, even not induced! Theorem (Kühn, Osthus 2004)

For every graph H and any s≥1, every graph of large average degree with no Ks,s subgraph contains an induced subdivision of H, where each edge is subdivided at least once.

G min. counter-ex. toχ(Ck)≤10¹⁰^k. G has large min. degree.

LetH =K_t,t =f(k)large. G contains an ind. subdiv of K_t.

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Forbidding C

₄

?

No induced cycle of length divisible byk In particular: noC4 subgraphs, even not induced!

For every graph H and any s≥1, every graph of large average degree with noK2,2 subgraph contains an induced subdivision of H, where each edge is subdivided at least once.

G min. counter-ex. toχ(Ck)≤10¹⁰^k.

G has large min. degree. LetH =K_t,t =f(k)large. G contains an ind. subdiv of K_t.

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Forbidding C

₄

?

For every graph H and any s≥1, every graph of large average degreewith noK2,2 subgraph contains an induced subdivision of H, where each edge is subdivided at least once.

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Forbidding C

₄

?

For every graph H and any s≥1, every graph of large average degreewith noK2,2 subgraph contains an induced subdivision of H, where each edge is subdivided at least once.

LetH=K_t,t =f(k)large.

G contains an ind. subdiv of K_t.

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Forbidding C

₄

?

For every graph H and any s≥1, every graph of large average degreewith noK2,2 subgraphcontains an induced subdivisionof H, where each edge is subdivided at least once.

LetH=K_t,t =f(k)large.

G contains an ind. subdiv of Kt.

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G has no triangle, no ind. C4, no hole of length divisible by k.

Theorem

Every graph with large average degree and no C₄ subgraph contains an ind. (≥1)-subdivision of Kt.

G

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Theorem

G

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Theorem

G

_K

t

3 1 2 5

k colors

⇒ Ramsey: ∃ monochr.

clique of sizek Kt

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The Result

LetC_3,2k≥6 be the class of graphs with no triangle and no hole of even length at least 6.

Theorem (L. 2015⁺)

There exists c >0such that for every graph G ∈ C_3,2k≥6, χ(G)≤c.

LetC_3,5,2k≥6 be the class of graphs with no triangle, noC5 and no hole of even length at least 6.

Lemma

There exists c⁰ >0 such that for every graph G ∈ C_3,5,2k≥6, χ(G)≤c⁰.

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The Result

LetC_3,2k≥6 be the class of graphs with no triangle and no hole of even length at least 6.

Theorem (L. 2015⁺)

There exists c >0such that for every graph G ∈ C_3,2k≥6, χ(G)≤c.

LetC_3,5,2k≥6 be the class of graphs with no triangle, noC5 and no hole of even length at least 6.

Lemma

There exists c⁰ >0 such that for every graph G∈ C_3,5,2k≥6, χ(G)≤c⁰.

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Parity Changing Path

H

G1 G2 G3

P1 P2 P3

y1 x2 y2 x3 y3 x4

x1

leftovers A Parity Changing Path (PCP) of order`is a sequence (G1,P₁, . . . ,G_`,P_`,H) such that:

There is an odd and an even path from x_i to y_i,∀i. Pi has length ≥2,∀i.

H is connected andχ(H) is the leftovers.

χ(Gi)≤4

x1 is theorigin of the PCP.

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Sketch of proof

1 Big χ⇒ Grow a PCP

2 Big χ⇒ Grow a rooted PCP in N_k

3 Having a neighbor inH ⇒ having neighbors everywhere

4 The active lift (∼ parents of the PCP) has bigχ.

5 Conclusion

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Sketch of proof

1 Big χ⇒ Grow a PCP

2 Big χ⇒ Grow a rooted PCP in Nk

3 Having a neighbor inH ⇒ having neighbors everywhere

4 The active lift (∼ parents of the PCP) has bigχ.

5 Conclusion

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Lemma

Let G ∈ C_3,5,2k≥6 be connected, v ∈V(G) and δ=χ(G). Then∃ a PCP of order 1 with origin v and leftovers≥h(δ) =δ/2−8.

G₁ P1 y1 x₁

leftovers x₂ H

=v

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k−2

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k−1 N_k−2

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k−1 N_k−2

Nk: bigχ

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

x y

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

x y x⁰

y⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z x

y x⁰

y⁰ P0

P₀⁰

connected

neighb. in{x, y, x⁰, y⁰}

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

neighb. in{x, y, x⁰, y⁰}

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

neighb. in{x, y, x⁰, y⁰} Case 1:zis adj. toxory

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

Case 1.1:zy⁰∈E

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

Case 1.1:zy⁰∈E

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

Case 1.1:zy⁰∈E

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Lemma

G₁ P1 y1 x₁

leftovers x₂ H

=v P1

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

z Z z₁ M₁

x y x⁰

y⁰ P0

P₀⁰

connected

Case 1.1:zy⁰∈E

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected

Case 1.1:zy⁰∈E

M1

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected

M1

Case 1.2:zy⁰∈/E

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected

neighb. in{x, y, x⁰, y⁰} Case 2:zis adj. neither toxnor toy

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected

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Lemma

G₁ P1 y1 x₁

leftovers x2 H

=v

P1 H

v

N0

N1

. . .

N_k⁰⊆Nk

N_k−1 N_k−2

Z z

z₁

M1

x y x⁰

y⁰ P0

P₀⁰

connected