  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.
