Circumferences of regular claw-free graphs

A known result obtained independently by Fan and Jung is that every 3-connected k-regular graph on n   vertices contains a cycle of length at least min{3k,n}min{3k,n}. This raises the question of how much can be said about the circumferences of 3-connected k-regular claw-free graphs. In this paper, we show that every 3-connected k-regular claw-free graph on n   vertices contains a cycle of length at least min{6k-17,n}min{6k-17,n}.