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

TL;DR

This paper proves that all 3-tough graphs that do not contain a specific forbidden subgraph are Hamiltonian, advancing understanding of Chvátal's toughness conjecture for this class.

Contribution

It establishes that 3-tough $(K_2 old 3K_1)$-free graphs are Hamiltonian, a new result in the study of toughness and Hamiltonicity.

Findings

All 3-tough $(K_2 old 3K_1)$-free graphs are Hamiltonian.

Supports Chve1tal's conjecture for this class of graphs.

Extends known classes where toughness implies Hamiltonicity.

Abstract

Chv\'{a}tal 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.