Bounds for chromatic number in terms of even-girth and booksize

The even-girth of any graph G is the smallest length of any even cycle in G. For any two integers t, k with 0 <= t <= k - 2, we denote the maximum number of cycles of length k such that each pair of cycles intersect in exactly a unique path of length t by b(t,k)(G). This parameter is called the (t, k)-booksize of G. In this paper we obtain some upper bounds for the chromatic and coloring numbers of graphs in terms of even-girth and booksize. We also prove some bounds for graphs which contain no cycle of length t where t is a small and fixed even integer. (C) 2010 Elsevier B.V. All rights reserved.