On (δ, χ)-Bounded Families of Graphs

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.