Hourglasses and Hamilton cycles in 4-connected claw-free graphs

We show that if G is a 4-connected claw-free graph in which every induced hourglass subgraph S contains two non-adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4-connected claw-free, hourglass-free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005

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