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


A class Γ of graphs is vertex Ramsey if for allH∈Γ there existsG∈Γ such that for all partitions of the vertices ofG into two parts, one of the parts contains an induced copy ofH. Forb (T,K) is the class of graphs that induce neitherT norK. LetT(k, r) be the tree with radiusr such that each nonleaf is adjacent tok vertices farther from the root than itself. Gyárfás conjectured that for all treesT and cliquesK, there exists an integerb such that for allG in Forb(T,K), the chromatic number ofG is at mostb. Gyárfás' conjecture implies a weaker conjecture of Sauer that for all treesT and cliquesK, Forb(T,K) is not vertex Ramsey. We use techniques developed for attacking Gyárfás' conjecture to prove that for allq, r and sufficiently largek, Forb(T(k,r),K q ) is not vertex Ramsey.

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Research partially supported by Office of Naval Research grant N00014-90-J-1206.

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Kierstead, H.A., Zhu, Y. Classes of graphs that exclude a tree and a clique and are not vertex ramsey. Combinatorica 16, 493–504 (1996).

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Mathematics Subject Classification (1991)

  • 05 C