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

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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 n196, 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 n1510 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{12,3p5} 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 HQ(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].

Keywords

Claw-free graph
Hamiltonian cycle
Minimum degree condition
Ryjáček's closure concept
Catlin's reduction method

Cited by (0)

1

Research is supported by Butler University Academic Grant (2014).

2

Research is supported by the Natural Science Funds of China (No. 11431037) and by Specialized Research Fund for the Doctoral Program of Higher Education (No. 20131101110048).