### 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.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 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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## Cite this

*Electronic Journal of Combinatorics*,

*18*(1).