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H 1 Colouring graphs with no odd holes, and other stories

Auteurs : Seymour, Paul D. (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... Gyarfas' conjecture on odd holes proof of Gyarfas' conjecture levelling spine parent rule cograph of jumps Gyarfas' conjecture on long holes tree-decomposition uncle trees

Résumé : The chromatic number $\chi(G)$ of a graph $G$ is always at least the size of its largest clique (denoted by $\omega(G)$), and there are graphs $G$ with $\omega(G)=2$ and $\chi(G)$ arbitrarily large.
On the other hand, the perfect graph theorem asserts that if neither $G$ nor its complement has an odd hole, then $\chi(G)=\omega(G)$ . (A "hole" is an induced cycle of length at least four, and "odd holes" are holes of odd length.) What happens in between?
With Alex Scott, we recently proved the following, a 1985 conjecture of Gyárfás:

For graphs $G$ with no odd hole, $\chi(G)$ is bounded by a function of $\omega(G)$.

Gyárfás also made the stronger conjecture that for every integer $k$ and for all graphs $G$ with no odd hole of length more than $k$, $\chi(G)$ is bounded by a function of $k$ and $\omega(G)=2$. This is far from settled, and indeed the following much weaker statement is not settled: for every integer $k$, every triangle-free graph with no hole of length at least $k$ has chromatic number bounded by a function of $k$. We give a partial result towards the latter:

For all $k$, every triangle-free graph with no hole of length at least $k$ admits a tree-decomposition into bags with chromatic number bounded by a function of $k$.

Both results have quite pretty proofs, which will more-or-less be given in full.

Codes MSC :
05C15 - Coloring of graphs and hypergraphs
05C35 - Extremal problems (graph theory)
05C85 - Graph algorithms

 Informations sur la Vidéo Langue : Anglais Date de publication : 04/02/15 Date de captation : 21/01/15 Sous collection : Research talks arXiv category : Combinatorics Domaine : Combinatorics Format : QuickTime (.mov) Durée : 00:55:58 Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-01-21_Seymour.mp4 Informations sur la rencontre Nom de la rencontre : International workshop on graph decomposition / Rencontre internationale sur les méthodes de décomposition de graphesOrganisateurs de la rencontre : Kreutzer, Stephan ; Paul, Christophe ; Trotignon, Nicolas ; Wollan, PaulDates : 19/01/15 - 23/01/15 Année de la rencontre : 2015 Citation Data DOI : 10.24350/CIRM.V.18671503 Cite this video as: Seymour, Paul D. (2015). Colouring graphs with no odd holes, and other stories. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18671503 URI : http://dx.doi.org/10.24350/CIRM.V.18671503

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