Closure operation for even factors on claw-free graphs

Abstract

Ryjáček (1997) [6] defined a powerful closure operation cl(G) on claw-free graphs G. Very recently, Ryjáček et al. (2010) [8] have developed the closure operation cl2f(G) on claw-free graphs which preserves the (non)-existence of a 2-factor. In this paper, we introduce a closure operation clse(G) on claw-free graphs that generalizes the above two closure operations. The closure of a graph is unique determined and the closure turns a claw-free graph into the line graph of a graph containing no cycle of length at most 5 and no cycles of length 6 satisfying a certain condition and no induced subgraph being isomorphic to the unique tree with a degree sequence 111133. We show that these closure operations on claw-free graphs all preserve the minimum number of components of an even factor. In particular, we show that a claw-free graph G has an even factor with at most k components if and only if clse(G) (cl(G),cl2f(G), respectively) has an even factor with at most k components. However, the closure operation does not preserve the (non)-existence of a 2-factor.