Hamiltonicity of 3-tough $(K_2 \cup 3K_1)$-free graphs

Abstract

Chvátal conjectured in 1973 the existence of some constant $t$ such that all $t$-tough graphs with at least three vertices are hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. We say that a graph is $(K_2 \cup 3K_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup 3K_1$, where $K_2 \cup 3K_1$ is the disjoint union of an edge and three isolated vertices. In this paper, we show that every 3-tough $(K_2 \cup 3K_1)$-free graph with at least three vertices is hamiltonian.