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

TL;DR

This paper proves that 7-tough graphs free of the disjoint union of a path of length 3 and three isolated vertices are Hamiltonian, confirming a conjecture for this class of graphs.

Contribution

It confirms the conjecture that 7-tough (P3 ∪ 3P1)-free graphs are Hamiltonian, extending previous results to a broader class of graphs.

Findings

7-tough (P3 ∪ 3P1)-free graphs are Hamiltonian

Confirms a conjecture for this graph class

Extends known results from (P3 ∪ 2P1)-free graphs

Abstract

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.