2-factors in Claw-free Graphs

We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced K1,3.) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ n−2 3 , G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ n−24 3 . We also show that this result is sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k.