Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs

We introduce a closure concept that turns a claw-free graph into the line graph of a multigraph while preserving its (non-)Hamiltonconnectedness. As an application, we show that every 7-connected claw-free graph is Hamilton-connected, and we show that the wellknown conjecture by Matthews and Sumner (every 4-connected clawfree graph is hamiltonian) is equivalent with the statement that every 4-connected claw-free graph is Hamilton-connected. Finally, we show a natural way to avoid the non-uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. 1 Notation and terminology In this paper, by a graph we mean a finite simple undirected graph G = (V (G), E(G)); whenever we allow multiple edges we say that G is a multigraph. For a vertex x ∈ V (G), dG(x) denotes the degree of x in G, NG(x) denotes the neighborhood of x in G (i.e. NG(x) = {y ∈ V (G)| xy ∈ E(G)}) and NG[x] denotes the closed neighborhood of x in G (i.e. NG[x] = NG(x) ∪ {x}). For x, y ∈ V (G), distG(x, y) denotes the distance of x, y in G. A universal vertex 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 [email protected], [email protected]. Research supported by grants No. 1M0545 and MSM 4977751301 of the Czech Ministry of Education.