On 2-factors with a bounded number of odd components

A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjáček, Saito, and Schelp (Closure, 2-factors, and cycle coverings in claw-free graphs, J. Graph Theory, 32 (1999), no. 2, 109-117) and show that the number of odd components of a 2-factor in a claw-free graph is stable under Ryjáček’s closure operation. We also consider conditions that ensure the existence of of a pair of disjoint 1-factors in a claw-free graph, as the union of such a pair is a 2-factor with no odd cycles.