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 + 5 10 , then either H is Hamiltonian, or the Ryjacek's closure c l ( 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 δ ( H ) ≥ n + 34 12 and conjectured that similar result would also hold even if δ ( H ) ≥ n + 12 13 . 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 , 3 p − 5 } such that for any 3-connected claw-free graph H of order n > N and with δ ( H ) ≥ n + ϵ p , 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] .