Hamilton-Connectedness in (K2 ∪ kK1)-Free Tough Graphs

The toughness of a non-complete graph [Formula: see text], denoted by [Formula: see text], is defined as the maximum real number [Formula: see text] such that [Formula: see text] for every cutset [Formula: see text], where [Formula: see text] denotes the number of components in [Formula: see text]. In this paper, we show that every [Formula: see text]-connected [Formula: see text]-free graph [Formula: see text] with [Formula: see text] is Hamilton-connected. As a direct corollary, every [Formula: see text]-tough [Formula: see text]-free graph is Hamilton-connected. This improves the results of Zheng et al. [Toughness, forbidden subgraphs, and Hamilton-connected graphs. Discuss. Math. Graph Theory, 42(1):187–196, 2021], which states that every [Formula: see text]-free graph [Formula: see text] with [Formula: see text] is Hamilton-connected.