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 GM 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) 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. Finally, by reconsidering the relation between degree conditions and clique coverings in the case of the strong closure GM , we prove that every 3-connected claw-free graph G of minimum degree δ(G) ≥ 24 and minimum degree sum σ8(G) ≥ n + 50 (or, as a corollary, of order n ≥ 142 and minimum degree δ(G) ≥ n+50 8 ) is Hamiltonconnected. We also show that our results are asymptotically sharp. 1 Notation and terminology In this paper we follow the most common graph-theoretic terminology and notation and for notations and concepts not defined here we refer the reader to [3]. Specifically, by a graph we mean a finite simple undirected graph G = (V (G), E(G)); whenever we allow multiedges (multiple edges), we say that G is a multigraph. By a multiedge in a multigraph we mean an induced subgraph X ⊂ G such that |V (X)| = 2 and |E(X)| ≥ 2. More precisely, for an edge e1e2, we can define the induced subgraph X ⊂ G with V (X) = {e1, e2} and say that e1e2 is a single edge (multiedge) if |E(X)| = 1 (|E(X)| ≥ 2), respectively. The number |E(X)| will be also called the multiplicity of the Department of Mathematics, University of West Bohemia, and Institute for Theoretical Computer Science (ITI), Charles University, P.O. Box 314, 306 14 Pilsen, Czech Republic, e-mail {rkuzel,ryjacek,teska,vranap}@kma.zcu.cz Research supported by grants No. 1M0545 and MSM 4977751301 of the Czech Ministry of Education.