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. 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) δ ( G ) ≥ 4 and G can be covered by at most 6 cliques, (iii) δ ( G ) ≥ 6 and G can be covered by at most 7 cliques. y, 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 δ ( G ) ≥ 24 and minimum degree sum σ 8 ( G ) ≥ n + 50 (or, as a corollary, of order n ≥ 142 and minimum degree δ ( G ) ≥ n + 50 8 ) is Hamilton-connected. o show that our results are asymptotically sharp.