Closure operation for even factors on claw-free graphs

Ryjáček (1997) [6] defined a powerful closure operation cl(G) on claw-free graphs GG. 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 GG has an even factor with at most kk components if and only if clse(G) (cl(G),cl2f(G), respectively) has an even factor with at most kk components. However, the closure operation does not preserve the (non)-existence of a 2-factor.