On lower bounds for the chromatic number in terms of vertex degree

In this paper we discuss the existence of lower bounds for the chromatic number of graphs in terms of the average degree or the coloring number of graphs. We obtain a lower bound for the chromatic number of K1,tK1,t-free graphs in terms of the maximum degree and show that the bound is tight. For any tree TT, we obtain a lower bound for the chromatic number of any K2,tK2,t-free and TT-free graph in terms of its average degree. This answers affirmatively a modified version of Problem 4.3 in [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995]. More generally, we discuss δδ-bounded families of graphs and then we obtain a necessary and sufficient condition for a family of graphs to be a δδ-bounded family in terms of its induced bipartite Turán number. Our last bound is in terms of forbidden induced even cycles in graphs; it extends a result in [S.E. Markossian, G.S. Gasparian, B.A. Reed, ββ-perfect graphs, J. Combin. Theory Ser. B 67 (1996) 1–11].