On 2-factors with long cycles in 3-connected claw-free graphs

Abstract

For a graph H, let δ(H) be the minimum degree and let c(H) be the length of a longest cycle in H. A 2-factor of H is a spanning subgraph of H in which every component is a cycle. In [Discrete Math. 313 (2013) 1934-1943], Čada and Chiba asked if H is a 3-connected claw-free graph of order n with δ(H) sufficiently large, does H have a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}? In this paper, to answer their question, we show that if H is a 3-connected claw-free graph of order n with δ(H)≥15, then H has a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}.