Minimum degree conditions for the Hamiltonicity of 3-connected claw-free graphs

Abstract

Settling a conjecture of Kuipers and Veldman posted in Favaron and Fraisse (2001) [9], Lai et al. (2006) [15] proved that if H is a 3-connected claw-free simple graph of order n≥196, and if δ(H)≥n+510, then either H is Hamiltonian, or the Ryjáček's closure cl(H)=L(G) where G is the graph obtained from the Petersen graph P by adding n−1510 pendant edges at each vertex of P. Recently, Li (2013) [17] improved this result for 3-connected claw-free graphs H with δ(H)≥n+3412 and conjectured that similar result would also hold even if δ(H)≥n+1213. In this paper, we show that for any given integer p>0 and real number ϵ, there exist an integer N=N(p,ϵ)>0 and a family Q(p), which can be generated by a finite number of graphs with order at most max⁡{12,3p−5} such that for any 3-connected claw-free graph H of order n>N and with δ(H)≥n+ϵp, H is Hamiltonian if and only if H∉Q(p).As applications, we improve both results in Lai et al. (2006) [15] and in Li (2013) [17], and give a counterexample to the conjecture in Li (2013) [17].