Closure operation for even factors on claw-free graphs

Ryjacek [Z. Ryjáček: On a closure concept in claw-free graphs. Journal of Combinatorial Theory Ser. B 70 (1997), 217-224] defined a powerful closure operation cl(G) on claw-free graphs G. Very recently, Ryjacek, Yoshimoto and the talker developed the closure operation cl (G) on claw-free graphs which preserves the (non)-existence of a 2-factor. In this talk, we introduce a closure operation cl(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 certain condition and no induced subgraph being isomorphic to the unique tree with the 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 cl(G) (cl(G), cl (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.