Hamiltonicity of claw-free graphs and Fan-type conditions

Abstract

For a graph H, let δF(H)=min{max{d(u),d(v)}|for any u,v∈V(H) with distH(u,v)=2}. For a given integer p>0 and a given number ϵ, a graph H of order n is said to satisfy a Fan-type condition if δF(H)≥2n+ϵp. In this paper, we show that like the planar graphs having obstruction set {K5,K3,3}, for the family of k-connected claw-free Hamiltonian graphs H with δF(H)≥2n+ϵp, if k=3 it has a finite obstruction set in which each graph has order at most max{9p∕2−20,10}, and if k=2 it has an obstruction set in which each graph has at most max{9p−20,0} vertices of degree greater than 3. We determine the best possible values of p and ϵ when the obstruction set is empty. We show that for k-connected claw-free simple graphs H of order n with n large enough, each of the following holds:(a) if k=2 and δF(H)≥2n−114 then H is Hamiltonian;(b) if k=3 and δF(H)≥2n−198 then H is Hamiltonian.These bounds are sharp. For k=3 and a given integer p, since the obstruction set is finite and can be determined in polynomial time, improvements to (b) for other given values of p are possible with the help of a computer.