Radius three trees in graphs with large chromatic number

Henry Kierstead, Yingxian Zhu

Research output: Contribution to journalArticle

17 Scopus citations

Abstract

A class γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T, the class Forb (T), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

Original languageEnglish (US)
Pages (from-to)571-581
Number of pages11
JournalSIAM Journal on Discrete Mathematics
Volume17
Issue number4
DOIs
StatePublished - Apr 1 2004

Keywords

  • Chromatic number
  • Clique number
  • Forbidden induced subgraph
  • Radius three tree
  • Template

ASJC Scopus subject areas

  • Mathematics(all)

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