Cliques, stable sets and colorings Nicolas Bousquet (joint work with Marthe Bonamy, Aur´ elie Lagoutte and St´ ephan Thomass´ e) 1/25
1 χ -bounded classes 2 Erd˝ os-Hajnal Erd˝ os-Hajnal and χ -boundedness Paths and antipaths Cycles and anticycles 3 Separate cliques and stable sets 4 Conclusion 2/25
First definitions • ω the maximum size of a clique. • α the maximum size of a stable set. • χ the chromatic number. • P k : induced path on k vertices. • C k : induced cycle on k vertices. • class = class closed under induced subgraphs. • n : number of vertices of the graph. 2/25
Chromatic number and stable sets Observation χ ≥ n α . A coloring is a partition of the vertex set into independent sets. 3/25
Chromatic number and stable sets Observation χ ≥ n α . A coloring is a partition of the vertex set into independent sets. 3/25
Chromatic number and stable sets Observation χ ≥ n α . A coloring is a partition of the vertex set into independent sets. At least n α colors are necessary since each color class has size at most α . 3/25
Chromatic number and stable sets Chromatic number at most c = Partition into c stable sets 4/25
Chromatic number and stable sets Chromatic number at most c = Partition into c stable sets ⇓ Fractional chromatic number number at most c ( ⇒ Existence of a stable set of size n c ). 4/25
Chromatic number and stable sets Chromatic number at most c = Partition into c stable sets ⇓ Fractional chromatic number number at most c ( ⇒ Existence of a stable set of size n c ). ⇓ n Existence of an empty bipartite graph of size 2 c . 4/25
Chromatic number and stable sets Chromatic number at most c = Partition into c stable sets ⇓ Fractional chromatic number number at most c ( ⇒ Existence of a stable set of size n c ). ⇓ n Existence of an empty bipartite graph of size 2 c . Question : Reverse of these implications ? • First implication : FALSE. • Second implication : we only have a polynomial clique or a polynomial stable set. 4/25
Chromatic number and cliques Observation : We always have ω ≤ χ . ⇒ Existence of a reverse function ? 5/25
Chromatic number and cliques Observation : We always have ω ≤ χ . ⇒ Existence of a reverse function ? Answer (Erd˝ os) NO ! Proof : Using the “probabilistic method” 5/25
Chromatic number and cliques Observation : We always have ω ≤ χ . ⇒ Existence of a reverse function ? Answer (Erd˝ os) NO ! Proof : Using the “probabilistic method” • Put every edge with probability p = n − 2 3 . n • For every k , the average size of a stable set is less than 2 k . • The average number of triangle is less than n 6 . 5/25
Chromatic number and cliques Observation : We always have ω ≤ χ . ⇒ Existence of a reverse function ? Answer (Erd˝ os) NO ! Proof : Using the “probabilistic method” • Put every edge with probability p = n − 2 3 . n • For every k , the average size of a stable set is less than 2 k . • The average number of triangle is less than n 6 . ⇒ After the deletion of n / 2 vertices there remain a triangle free graph with small stable sets. 5/25
Chromatic number and cliques Observation : We always have ω ≤ χ . ⇒ Existence of a reverse function ? Answer (Erd˝ os) NO ! Proof : Using the “probabilistic method” • Put every edge with probability p = n − 2 3 . n • For every k , the average size of a stable set is less than 2 k . • The average number of triangle is less than n 6 . ⇒ After the deletion of n / 2 vertices there remain a triangle free graph with small stable sets. Definition ( χ -bounded) A class is χ -bounded if χ ≤ f ( ω ). Example : Graphs with no P k are χ -bounded (Gy´ arf´ as ’87). 5/25
Gyarf´ as proof (for triangle-free graphs) Take a vertex u . • A connected component X of G \ N ( u ) has chromatic number at least χ − 1. χ − 1 v u 6/25
Gyarf´ as proof (for triangle-free graphs) Take a vertex u . • A connected component X of G \ N ( u ) has chromatic number at least χ − 1. • Take v a vertex of N ( u ) with a neighbor in X . χ − 1 v u 6/25
Gyarf´ as proof (for triangle-free graphs) Take a vertex u . • A connected component X of G \ N ( u ) has chromatic number at least χ − 1. • Take v a vertex of N ( u ) with a neighbor in X . • Restrict the graph to v ∪ X and repeat. w χ − 2 v u 6/25
Gyarf´ as proof (for triangle-free graphs) Take a vertex u . • A connected component X of G \ N ( u ) has chromatic number at least χ − 1. • Take v a vertex of N ( u ) with a neighbor in X . • Restrict the graph to v ∪ X and repeat. w χ − 2 v u When the clique is unbounded, the function becomes exponential... 6/25
χ -bounded classes • P k -free graphs • Star-free graphs • Disk graphs are χ -bounded. • Perfect graphs 7/25
χ -bounded classes • P k -free graphs • Star-free graphs • Disk graphs are χ -bounded. • Perfect graphs But for many classes we do not know if they are χ -bounded or not. • Long hole-free graphs. • Odd cycle-free graphs. • Wheel-free graphs. 7/25
χ -bounded classes • P k -free graphs • Star-free graphs • Disk graphs are χ -bounded. • Perfect graphs But for many classes we do not know if they are χ -bounded or not. • Long hole-free graphs. • Odd cycle-free graphs. • Wheel-free graphs. For χ -bounded classes of graphs, we try to find the best possible function f . Conjecture (Gy´ arf´ as ’87) A graph with no copy of P k has chromatic number at most Poly ( k , ω ). 7/25
1 χ -bounded classes 2 Erd˝ os-Hajnal Erd˝ os-Hajnal and χ -boundedness Paths and antipaths Cycles and anticycles 3 Separate cliques and stable sets 4 Conclusion 8/25
Erd˝ os-Hajnal and χ -boundedness Conjecture (Erd˝ os Hajnal ’89) A graph with no copy of P k has a clique or a stable set of size n ǫ . Folklore If a class C of graphs satisfies χ ≤ ω c then C has a polynomial clique or stable set. 8/25
Erd˝ os-Hajnal and χ -boundedness Conjecture (Erd˝ os Hajnal ’89) A graph with no copy of P k has a clique or a stable set of size n ǫ . Folklore If a class C of graphs satisfies χ ≤ ω c then C has a polynomial clique or stable set. Proof : 1 2 c ⇒ OK. • Either ω ≥ n 2 c ⇒ χ ≤ √ n . 1 • Or ω ≤ n So there is a stable set of size √ n . ⇒ Polynomial χ -bounded stronger than Erd˝ os-Hajnal. 8/25
Erd˝ os-Hajnal conjecture What is the value of max( ω, α ) if some graph H is forbidden ? α = n α ≥ √ n log n α or ω are at least √ n α or ω are at least √ n 9/25
Erd˝ os-Hajnal conjecture What is the value of max( ω, α ) if some graph H is forbidden ? α = n α ≥ √ n log n α or ω are at least √ n α or ω are at least √ n Conjecture (Erd˝ os-Hajnal ’89) For every H , there exists ǫ > 0 such that every H -free graph satisfies max( α, ω ) ≥ n ǫ . 9/25
On the importance of H Lemma (Grimmet, Mc Diarmid ’75) Random graphs satisfy α, ω = O (log n ). 10/25
On the importance of H Lemma (Grimmet, Mc Diarmid ’75) Random graphs satisfy α, ω = O (log n ). Sketch of proof : 2 ) 2 log 2 n Probability that a set of size 2 log n is a clique ≈ ( 1 Number of such sets ≈ n 2 log n = 2 2 log 2 n . ⇒ Average number of cliques ≈ 1. 10/25
On the importance of H Lemma (Grimmet, Mc Diarmid ’75) Random graphs satisfy α, ω = O (log n ). Sketch of proof : 2 ) 2 log 2 n Probability that a set of size 2 log n is a clique ≈ ( 1 Number of such sets ≈ n 2 log n = 2 2 log 2 n . ⇒ Average number of cliques ≈ 1. Lemma (Grimmet, Mc Diarmid ’75) n Random graphs satisfy χ = O ( log n ). 10/25
Prime graphs Theorem (Alon, Pach, Solymosi) If the Erd˝ os-Hajnal conjecture holds for every prime graph H , then it holds for every graph. 11/25
Prime graphs Theorem (Alon, Pach, Solymosi) If the Erd˝ os-Hajnal conjecture holds for every prime graph H , then it holds for every graph. Interesting prime graphs on 4 vertices : P 4 . � 11/25
Prime graphs Theorem (Alon, Pach, Solymosi) If the Erd˝ os-Hajnal conjecture holds for every prime graph H , then it holds for every graph. Interesting prime graphs on 4 vertices : P 4 . � Interesting prime graphs on 5 vertices : bull, P 5 , C 5 and their complements. • Bull : Chudnovsky, Safra ’08. � • P 5 , C 5 : widely open. 11/25
Prime graphs Theorem (Alon, Pach, Solymosi) If the Erd˝ os-Hajnal conjecture holds for every prime graph H , then it holds for every graph. Interesting prime graphs on 4 vertices : P 4 . � Interesting prime graphs on 5 vertices : bull, P 5 , C 5 and their complements. • Bull : Chudnovsky, Safra ’08. � • P 5 , C 5 : widely open. ⇒ What happens if we enforce stronger conditions... Idea : forbid a graph and its complement. 11/25
Erd˝ os-Hajnal for paths and antipaths Theorem (Chudnovsky, Zwols ’11) Graphs with no P 5 nor complement of P 6 have the Erd˝ os-Hajnal property. Theorem (Chudnovsky, Seymour ’12) Graphs with no P 5 nor complement of P 7 have the Erd˝ os-Hajnal property. 12/25
Erd˝ os-Hajnal for paths and antipaths Theorem (B., Lagoutte, Thomass´ e ’13) Graphs with no P k nor its complement have the Erd˝ os-Hajnal property. 13/25
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