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.