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SUMMARY:Paul Seymour\, Polynomial bounds for chromatic number
DESCRIPTION:The Gyárfás-Sumner conjecture says that for every forest $H$\, there is a function $f$ such that the chromatic number $\chi(G)$ is at most $f(\omega(G))$ for every $H$-free graph $G$ (“$H$-free” means with no induced subgraph isomorphic to $H$\, and $\omega(G)$ is the size of the largest clique of $G$). This well-known conjecture has been proved only for a few types of forest. \nNevertheless\, there is a much stronger conjecture\, due to Esperet: that for every forest $H$\, there is a polynomial function $f$ as above. As one might expect\, this has been proved for even fewer types of forest; and the smallest tree $H$ for which Esperet’s conjecture is not known is the five-vertex path $P_5$. \nA third notorious conjecture is the Erdős-Hajnal conjecture\, that for every graph $H$\, there exists $c>0$ such that $\alpha(G)\omega(G)\ge |G|^c$ for every $H$-free graph $G$ (where $\alpha(G)$ is the size of the largest stable set of $G$). The smallest graph $H$ for which this is not known is also $P_5$\, which\, conveniently\, is a forest; and every forest that satisfies Esperet’s conjecture also satisfies the Erdős-Hajnal conjecture. So there is substantial interest in the chromatic numbers of $P_5$-free graphs. The best upper bound that was previously known\, due to Esperet\, Lemoine\, Maffray\, and Morel\, was that $\chi(G)\le (5/27)3^\omega(G)$ for every $P_5$-free graph $G$ with $\omega(G) > 2$. In recent work with Alex Scott and Sophie Spirkl\, we have proved several results related to Esperet’s conjecture\, including proofs of its truth for some new types of forest $H$\, and a “near-polynomial” bound when $H = P_5$\, that $\chi(G) \le \omega(G)^{\log_2(\omega(G))}$ for every $P_5$-free graph $G$ with $\omega(G) > 2$. We survey these results and give a proof of the new bound for $P_5$.
URL:https://dimag.ibs.re.kr/event/2021-10-08/
LOCATION:Zoom ID: 869 4632 6610 (ibsdimag)
CATEGORIES:Virtual Discrete Math Colloquium
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