Extending the Gyárfás-Sumner conjecture

Research output: Contribution to journalArticle

4 Scopus citations

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 Gyárfás 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 H1 that is a disjoint union of cliques, and every complete multipartite graph H2, there exists k such that every graph containing neither of H1, H2 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 Gyárfás [5] and for which we give a short proof.

Original languageEnglish (US)
Pages (from-to)11-16
Number of pages6
JournalJournal of Combinatorial Theory. Series B
Volume105
Issue number1
DOIs
StatePublished - Jan 1 2014

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Induced
  • Ramsey
  • Subgraph

Fingerprint Dive into the research topics of 'Extending the Gyárfás-Sumner conjecture'. Together they form a unique fingerprint.

  • Cite this