Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs

TL;DR

This paper proves that all 7-tough $(P_3rac12; 2P_1)$-free graphs with at least three vertices are Hamiltonian, confirming Chve1tal's toughness conjecture for this class of graphs.

Contribution

The paper establishes that 7-toughness suffices for Hamiltonicity in $(P_3rac12; 2P_1)$-free graphs, verifying the conjecture within this graph class.

Findings

Every 7-tough $(P_3rac12; 2P_1)$-free graph is Hamiltonian.

The result applies to graphs with at least three vertices.

Supports Chve1tal's toughness conjecture for this class.

Abstract

The toughness of a noncomplete graph $G$ is the maximum real number $t$ such that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$ for every cutset $S$ of $G$, and the toughness of a complete graph is defined to be $\infty$. Determining the toughness for a given graph is NP-hard. Chv\'{a}tal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chv\'{a}tal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.