Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs

The toughness of a noncomplete graph G is the maximum real number t such that the ratio of |S| to the number of components of G−S is at least t for every cutset S of G, and the toughness of a complete graph is defined to be ∞. Determining the toughness for a given graph is NP-hard. Chvátal’s toughness conjecture, stating that there exists a constant t0 such that every graph with toughness at least t0 is hamiltonian, is still open for general graphs. A graph is called (P3 ∪ 2P1)-free if it does not contain any induced subgraph isomorphic to P3 ∪ 2P1, the disjoint union of P3 and two isolated vertices. In this paper, we confirm Chvátal’s toughness conjecture for (P3 ∪ 2P1)-free graphs by showing that every 7-tough (P3 ∪ 2P1)-free graph on at least three vertices is hamiltonian.

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