• 1-Hamilton-connected;
  • claw-free graph;
  • closure;
  • line graph;
  • Thomassen Conjecture


A graph G is 1-Hamilton-connected if inline image is Hamilton-connected for every vertex inline image. In the article, we introduce a closure concept for 1-Hamilton-connectedness in claw-free graphs. If inline image is a (new) closure of a claw-free graph G, then inline image is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, inline image is the line graph of a multigraph, and for some inline image, inline image is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassen's Conjecture (every 4-connected line graph is hamiltonian) is equivalent to the statement that every 4-connected claw-free graph is 1-Hamilton-connected, and we present results showing that every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected and that every 4-connected claw-free and hourglass-free graph is 1-Hamilton-connected.