Hamilton-connected claw-free graphs with Ore-degree conditions

Abstract

Let H be a 3-connected claw-free graph on n vertices, and define σ2(H)=min{dH(u)+dH(v):uv∉E(H)}. Kužel et al. proved that H is Hamilton-connected if n≥142 and δ(H)≥n+508. Let HU be an UM-closure of H. We generalize the result above with degree sum conditions by proving the following.(i) For any positive integer p and real number ϵ, there exist an integer N(p,ϵ)=8p2−(2p+1)|ϵ|−2p>0 and a family Gp(r), which can be generated by a finite number of graphs of order r≤max{8,6p+3}, such that if n>N(p,ϵ) and σ2(H)≥n+ϵp(or δ(H)≥n+ϵ2p), then H is Hamilton-connected if and only if HU=L(G) for G∉Gp(r).(ii) If n≥85 and σ2(H)≥n+44(or δ(H)≥n+48), then either H is Hamilton-connected or HU=L(V8′), where V8′ is the graph obtained from the Wagner graph V8 by attaching |E(V8′)|−128 pendant edges at each vertex of V8.