Every 4-connected {claw, hourglass}-free graph is 2-Hamiltonian

Abstract

A graph G is k-hamiltonian if the graph G - X is hamiltonian for any set X of k vertices of G. The claw is the complete bipartite graph K(1,3), and the hourglass is the unique graph with degree sequence 4,2,2,2,2. We show that every 4-connected {claw, hourglass}-free graph is 2-hamiltonian. The result can be easily extended to k-hamiltonicity, implying that, for k >= 2, a {claw, hourglass}-free graph G is k-hamiltonian if and only if G is (k+2)connected. This immediately implies that k-hamiltonicity is, for k >= 2, polynomial-time decidable in the class of {claw, hourglass}-free graphs.