Classes of graphs that exclude a tree and a clique and are not vertex Ramsey

Henry Kierstead, Yingxian Zhu

Research output: Contribution to journalArticle

Abstract

A class Γ of graphs is vertex Ramsey if for all H ∈ Γ there exists G ∈ Γ such that for all partitions of the vertices of G into two parts, one of the parts contains an induced copy of H. Forb (T,K) is the class of graphs that induce neither T nor K. Let T(k,r) be the tree with radius r such that each nonleaf is adjacent to k vertices farther from the root than itself. Gyárfás conjectured that for all trees T and cliques K, there exists an integer b such that for all G in Forb (T,K), the chromatic number of G is at most b. Gyárfás' conjecture implies a weaker conjecture of Sauer that for all trees T and cliques K, Forb (T,K) is not vertex Ramsey. We use techniques developed for attacking Gyárfás' conjecture to prove that for all q, r and sufficiently large k, Forb (T(k,r),Kq) is not vertex Ramsey.

Original languageEnglish (US)
Pages (from-to)493-504
Number of pages12
JournalCombinatorica
Volume16
Issue number4
DOIs
StatePublished - Jan 1 1996

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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