Two-factors with few cycles in claw-free graphs

AbstractLet G be a graph of order n. Define fk(G) (Fk(G)) to be the minimum (maximum) number of components in a k-factor of G. For convenience, we will say that fk(G)=0 if G does not contain a k-factor. It is known that if G is a claw-free graph with sufficiently high minimum degree and proper order parity, then G contains a k-factor. In this paper we show that f2(G)⩽n/δ for n and δ sufficiently large and G claw-free. In addition, we consider F2(G) for claw-free graphs and look at the potential range for the number of cycles in a 2-factor