Closure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs

We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and fourth authors, such that the strong closure G(M) of a claw-free graph G is the line graph of a multigraph containing at most two triangles or at most one double edge. Using the concept of strong closure, we prove that a 3-connected claw-free graph G is Hamilton-connected if G satisfies one of the following: (i) G can be covered by at most 5 cliques, (ii)delta(G) >= 4 and G can be covered by at most 6 cliques, (iii)delta(G) >= 6 and G can be covered by at most 7 cliques. Finally, by reconsidering the relation between degree conditions and clique coverings in the case of the strong closure G(M) we prove that every 3-connected claw-free graph G of minimum degree delta(G) >= 24 and minimum degree sum sigma(8)(G) >= n + 50 (or, as a corollary, of order n >= 142 and minimum degree delta(C) >= n+50/8) is Hamilton-connected. We also show that our results are asymptotically sharp. (C) 2012 Elsevier B.V. All rights reserved.