On (δ, Χ)-bounded families of graphs

András Gyárfás, Manouchehr Zaker

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A family F of graphs is said to be (δ,Χ)-bounded if there exists a function f(x) satisfying f(x) → ∞ as x → ∞, such that for any graph G from the family, one has f(δ(G)) ≤ Χ(G), where δ(G) and Χ(G) denotes the minimum degree and chromatic number of G, respectively. Also for any set {H1,H2,... Hk} of graphs by Forb(H1,H2,... Hk) we mean the class of graphs that contain no Hi as an induced subgraph for any i = 1,k. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree T and positive integer ℓ, Forb(T,Kℓ,ℓ) is a (δ,Χ)-bounded family. Then we obtain a necessary and sufficient condition for Forb(H1,H2,..., Hk) to be a (δ,Χ)-bounded family, where {H1,H2,. Hk} is any given set of graphs. Next we study (δ,Χ)-boundedness of Forb(C) where C is an infinite collection of graphs. We show that for any positive integer ℓ, Forb(Kℓ,ℓ,C6,C8,) is (δ,Χ)-bounded. Finally we show a similar result when C is a collection consisting of unicyclic graphs.

Original languageEnglish
JournalElectronic Journal of Combinatorics
Issue number1
Publication statusPublished - May 30 2011

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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