# Hamiltonian cycles in tough $(P_2\cup P_3)$-free graphs

## TL;DR

This paper proves that every 15-tough $(P_2rac12;P_3)$-free graph with at least three vertices contains a Hamiltonian cycle, advancing understanding of toughness conditions for Hamiltonicity.

## Contribution

It establishes a new toughness threshold for Hamiltonicity specifically in $(P_2rac12;P_3)$-free graphs, a class previously not fully characterized.

## Key findings

- 15-tough $(P_2rac12;P_3)$-free graphs are Hamiltonian
- Provides a specific toughness bound for Hamiltonicity in this graph class
- Advances the toughness-Hamiltonicity conjecture in restricted graph classes

## Abstract

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, respectively. In this paper, we show that every 15-tough $(P_2\cup P_3)$-free graph with at least three vertices is hamiltonian.