Hamiltonian cycles in 2-tough $2K_2$-free graphs

TL;DR

This paper proves that every 2-tough $2K_2$-free graph with at least three vertices contains a Hamiltonian cycle, confirming a conjecture and improving previous toughness bounds for Hamiltonicity.

Contribution

It establishes the Hamiltonicity of 2-tough $2K_2$-free graphs, lowering the toughness requirement from 3 to 2, which was previously conjectured.

Findings

Every 2-tough $2K_2$-free graph on at least three vertices is Hamiltonian.

Confirmed the conjecture by Gao and Pasechnik.

Improved the toughness bound for Hamiltonicity in this class of graphs.

Abstract

A graph $G$ is called a $2K_2$-free graph if it does not contain $2K_2$ as an induced subgraph. In 2014, Broersma, Patel and Pyatkin showed that every 25-tough $2K_2$-free graph on at least three vertices is Hamiltonian. Recently, Shan improved this result by showing that 3-tough is sufficient instead of 25-tough. In this paper, we show that every 2-tough $2K_2$-free graph on at least three vertices is Hamiltonian, which was conjectured by Gao and Pasechnik.