Closure and Hamilton-Connected Claw-Free Hourglass-Free Graphs

Abstract

The closure $\mathrm{cl}(G)$ of a claw-free graph $G$ is the graph obtained from $G$ by a series of local completions at eligible vertices, as long as this is possible. The construction of an SM-closure of $G$ follows the same operations, but if $G$ is not Hamilton-connected, then the construction terminates once every local completion at an eligible vertex leads to a Hamilton-connected graph. Although [see e.g. Ryjáček and Vrána (J Graph Theory 66:137–151, 2011)] $\mathrm{cl}(G)$ may be Hamilton-connected even if $G$ is not, we show that if $G$ is a 2-connected claw-free graph with minimum degree at least 3 such that its SM-closure is hourglass-free, then $G$ is Hamilton-connected if and only if the closure $\mathrm{cl}(G)$ of $G$ is Hamilton-connected.