On the Erd\H{o}s-Gy\'arf\'as conjecture in claw-free graphs

The Erd\H{o}s-Gy'{a}rf'{a}s conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erd\H{o}s-Gy'{a}rf'{a}s conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.