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

The closure $$\mathrm{cl}(G)$$cl(G) of a claw-free graph $$G$$G is the graph obtained from $$G$$G by a series of local completions at eligible vertices, as long as this is possible. The construction of an SM-closure of $$G$$G follows the same operations, but if $$G$$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)] $$\mathrm{cl}(G)$$cl(G) may be Hamilton-connected even if $$G$$G is not, we show that if $$G$$G is a 2-connected claw-free graph with minimum degree at least 3 such that its SM-closure is hourglass-free, then $$G$$G is Hamilton-connected if and only if the closure $$\mathrm{cl}(G)$$cl(G) of $$G$$G is Hamilton-connected.