New bounds for the chromatic number of graphs

In this article we first give an upper bound for the chromatic number of a graph in terms of its degrees. This bound generalizes and modifies the bound given in [11]. Next, we obtain an upper bound of the order of magnitude O(n(1-epsilon)) for the coloring number of a graph with small K-2,K-t (as subgraph), where n is the order of the graph. Finally, we give some bounds for chromatic number in terms of girth and book size. These bounds improve the best known bound, in terms of order and girth, for the chromatic number of a graph when its girth is an even integer. (C) 2008 Wiley Periodicals, Inc.