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Extending the Gyarfas-Sumner conjecture

Author(s): Chudnovsky, Maria; Seymour, Paul D

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Abstract: Say a set H of graphs is heroic if there exists k such that every graph containing no member of H as an induced subgraph has cochromatic number at most k. (The cochromatic number of G is the minimum number of stable sets and cliques with union V (G).) Assuming an old conjecture of Gyarfas and Sumner, we give a complete characterization of the finite heroic sets. This is a consequence of the following. Say a graph is k-split if its vertex set is the union of two sets A, B, where A has clique number at most k and B has stability number at most k. For every graph H-1 that is a disjoint union of cliques, and every complete multipartite graph H-2, there exists k such that every graph containing neither of H-1, H-2 as an induced subgraph is k-split. This in turn is a consequence of a bound on the maximum number of vertices in any graph that is minimal not k-split, a result first proved by Gyarfas [5] and for which we give a short proof. (C) 2013 Elsevier Inc. All rights reserved.
Publication Date: Mar-2014
Electronic Publication Date: 7-Dec-2013
Citation: Chudnovsky, Maria, Seymour, Paul. (2014). Extending the Gyarfas-Sumner conjecture. JOURNAL OF COMBINATORIAL THEORY SERIES B, 105 (11 - 16. doi:10.1016/j.jctb.2013.11.002
DOI: doi:10.1016/j.jctb.2013.11.002
ISSN: 0095-8956
EISSN: 1096-0902
Pages: 11 - 16
Type of Material: Journal Article
Version: Author's manuscript

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