Hamiltonian claw-free graphs involving minimum degrees

Favaron and Fraisse proved that any 3-connected claw-free graph HH with order nn 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 HH is a 3-connected claw-free graph of order n≥196n≥196, and if δ(H)≥n+610, then HH 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 HH is a 3-connected claw-free graph of order n≥363n≥363, and if δ(H)≥n+3412, then either HH is hamiltonian, or the Ryjác˘ek’s closure cl(H)cl(H) of HH is the line graph of one of the graphs obtained from the Petersen graph P10P10 by adding at least one pendant edge at each vertex vivi of P10P10 or by replacing exactly one vertex vivi of P10P10 with K̄2,p(p≥2) and adding at least one pendant edge at all other nine vertices vj∉V−{vi}vj∉V−{vi} of P10P10, and then by subdividing mm edges of P10P10 for m=0,1,2,…,15m=0,1,2,…,15, where K̄2,p is a connected bipartite graph.