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

Let H be a 3-connected claw-free graph on n vertices, and define σ 2 (H) = min { d H (u) + d H (v) : u v ∉ E (H) }. Kužel et al. proved that H is Hamilton-connected if n ≥ 142 and δ (H) ≥ n + 50 8 . Let H U 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) = 8 p 2 − (2 p + 1) | ϵ | − 2 p > 0 and a family G p (r) , which can be generated by a finite number of graphs of order r ≤ max { 8 , 6 p + 3 } , such that if n > N (p , ϵ) and σ 2 (H) ≥ n + ϵ p (or δ (H) ≥ n + ϵ 2 p ), then H is Hamilton-connected if and only if H U = L (G) for G ∉ G p (r). (i i) If n ≥ 85 and σ 2 (H) ≥ n + 4 4 (or δ (H) ≥ n + 4 8 ), then either H is Hamilton-connected or H U = L (V 8 ′) , where V 8 ′ is the graph obtained from the Wagner graph V 8 by attaching | E (V 8 ′) | − 12 8 pendant edges at each vertex of V 8. [ABSTRACT FROM AUTHOR]