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

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 delta(H) >= n+5/10, then either H is Hamiltonian, or the Ryjacek's closure cl(H) = L(G) where G is the graph obtained from the Petersen graph P by adding n-15/10 pendant edges at each vertex of P. Recently, Li (2013) [17] improved this result for 3-connected claw-free graphs H with delta(H) >= n+34/12 and conjectured that similar result would also hold even if delta(H) >= n+12/13 . In this paper, we show that for any given integer p > 0 and real number epsilon, there exist an integer N = N(p, is an element of) > 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 delta(H) >= n+ is an element of/p, H is Hamiltonian if and only if H is not an element of 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]. (C) 2016 Elsevier Inc. All rights reserved.