Maximal cycles in graphs of large girth

Abstract

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 $g\ge 4$ with $n$ vertices and at least $m\ge n$ 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.