Hamilton cycles in tough $(2P_2 \cup P_1)$-free graphs

TL;DR

This paper proves that 11-tough graphs that do not contain the specific subgraph $(2P_2 old P_1)$ are guaranteed to have a Hamiltonian cycle, advancing understanding of toughness conditions for Hamiltonicity.

Contribution

It establishes that 11-tough $(2P_2 old P_1)$-free graphs are Hamiltonian, providing a new partial result related to Chvtal's toughness conjecture.

Findings

11-tough $(2P_2 old P_1)$-free graphs are Hamiltonian

Supports the conjecture that higher toughness ensures Hamiltonicity in specific graph classes

Advances the classification of graph classes satisfying Chvtal's conjecture

Abstract

In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. While this conjecture is still open, work has been done to confirm it for several graph classes, including all $F$-free graphs for every 5-vertex linear forest $F$ other than $P_5$ and $2P_2\cup P_1$. In this note, we show that 11-tough $(2P_2 \cup P_1)$-free graphs on at least three vertices are Hamiltonian.