Circumferences of 2-factors in claw-free graphs

A graph G is said to be claw-free if G has no induced subgraph isomorphic to K 1 , 3 . Matthews and Sumner [M.M. Matthews, D.P. Sumner, Longest paths and cycles in K 1 , 3 -free graphs, J. Graph Theory 9 (1985) 269–277] proved that c ( G ) ≥ min { 2 δ ( G ) + 4 , n } if G is a 2-connected claw-free graph of order n , where for a graph G , let δ ( G ) and c ( G ) denote the minimum degree and the length of a longest cycle of G , respectively. In this paper, we extend this result for a graph G with δ ( G ) ≥ 7 as follows: if G is a 2-connected claw-free graph of order n with δ ( G ) ≥ 7 , then G has a 2-factor F such that c ( F ) ≥ min { 2 δ ( G ) + 4 , n } .