Hamiltonian claw-free graphs involving minimum degrees

• Authors: Mingchu Li
• Venue: Discrete Applied Mathematics
• Year: 2013

Favaron and Fraisse proved that any 3-connected claw-free graph H with order n and minimum degree @d(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 @d(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 @d(H)=n+3412, then either H is hamiltonian, or the Ryjac@?ek's closure cl(H) of H is the line graph of one of the graphs obtained from the Petersen graph P"1"0 by adding at least one pendant edge at each vertex v"i of P"1"0 or by replacing exactly one vertex v"i of P"1"0 with K@?"2","p(p=2) and adding at least one pendant edge at all other nine vertices v"j@?V-{v"i} of P"1"0, and then by subdividing m edges of P"1"0 for m=0,1,2,...,15, where K@?"2","p is a connected bipartite graph.