Hamiltonian claw-free graphs involving minimum degrees

Abstract

Favaron and Fraisse proved that any 3-connected claw-free graph H with order n and minimum degree δ(H)≥n+3810 is hamiltonian [O. Favaron and P. Fraisse, Hamiltonicity and minimum degree in 3-connected claw-free graphs, J. Combin. Theory B 82 (2001) 297–305]. Lai, Shao and Zhan showed that if H is a 3-connected claw-free graph of order n≥196, and if δ(H)≥n+610, then H is hamiltonian [H.-J. Lai, Y. Shao and M. Zhan, Hamiltonicity in 3-connected claw-free graphs, J. Combin. Theory B 96 (2006) 493–504]. In this paper, we improve the two results above and prove that if H is a 3-connected claw-free graph of order n≥363, and if δ(H)≥n+3412, then either H is hamiltonian, or the Ryjác˘ek’s closure cl(H) of H is the line graph of one of the graphs obtained from the Petersen graph P10 by adding at least one pendant edge at each vertex vi of P10 or by replacing exactly one vertex vi of P10 with K̄2,p(p≥2) and adding at least one pendant edge at all other nine vertices vj∉V−{vi} of P10, and then by subdividing m edges of P10 for m=0,1,2,…,15, where K̄2,p is a connected bipartite graph.