Hamiltonian cycles in $15$-tough ($P_3\cup 3P_1$)-free graphs

Abstract

A graph $G$ is called $t$-tough if $\left|S\right|\ge t\cdot w\left(G-S\right)$ for every cutset $S$ of $G$. Chvátal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph has a hamiltonian cycle. Gao and Shan have proved that every $7$-tough $(P_3\cup 2P_1)$-free grah is hamiltonian. In this paper, we confirm this conjecture for $(P_3\cup 3P_1)$-free graphs.