A note on hamiltonian cycles in 4-tough (P2 ∪ kP1)-free graphs

Abstract

Let t>0 be a real number and let G be a graph. We say G is t-tough if for every cutset S of G, the ratio of |S| to the number of components of G−S is at least t. The Toughness Conjecture of Chvátal, stating that there exists a constant t0 such that every t0-tough graph with at least three vertices is hamiltonian, is still open in general. For any given integer k≥1, a graph G is (P2∪kP1) free if G does not contain the disjoint union of P2 and k isolated vertices as an induced subgraph. In this note, we show that every 4-tough and 2k-connected (P2∪kP1)-free graph with at least three vertices is hamiltonian. This result in some sense is an “extension” of the classical Chvátal-Erdős Theorem that every max⁡{2,k}-connected (k+1)P1-free graph on at least three vertices is hamiltonian.