Hamiltonian cycles in $ 15 $-tough ($ P_{3}\cup 3P_{1} $)-free graphs

A graph $G$ is called $t$-tough if $\left|S\right|\geq t\cdot w\left(G-S\right)$ for every cutset $S$ of $G$. Chv\'atal 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.