Maximal cycles in graphs of large girth

In this paper we give a lower bound of the circumference of a graph in terms of girth and the number of edges. It is shown that a graph of girth at least g ≥ 4 with n vertices and at least m ≥ n edges contains a cycle of length at least (g - 2)m/(n - 2). In particular, for the case g = 4, an analogous result for 2-edge-connected weighted graphs is given. Moreover, the extremal case is characterized in both results.