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

AbstractThe 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 bt,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