Gyárfás, András and Zaker, M (2011) On (δ, Χ)-bounded families of graphs. ELECTRONIC JOURNAL OF COMBINATORICS, 18 (1). ISSN 1077-8926

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Abstract

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 {H 1,H 2,... H k} of graphs by Forb(H 1,H 2,... H k) we mean the class of graphs that contain no H i 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,K ℓ,ℓ) is a (δ,Χ)-bounded family. Then we obtain a necessary and sufficient condition for Forb(H 1,H 2,..., H k) to be a (δ,Χ)-bounded family, where {H 1,H 2,. H k} 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(K ℓ,ℓ,C 6,C 8,) is (δ,Χ)-bounded. Finally we show a similar result when C is a collection consisting of unicyclic graphs.

Item Type:	ISI Article
Subjects:	Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány
Depositing User:	EPrints Admin
Date Deposited:	05 Feb 2014 12:19
Last Modified:	05 Feb 2014 15:50
URI:	https://eprints.sztaki.hu/id/eprint/7601

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