The Chv\'atal-Erd\H{o}s condition for prism-Hamiltonicity

TL;DR

This paper proves that if a graph's independence number is at most twice its connectivity, then its prism is Hamiltonian, establishing a new sufficient condition for prism-Hamiltonicity.

Contribution

It confirms that the bound  \, ext{alpha}(G) \, extless= 2 \, ext{kappa}(G) guarantees prism-Hamiltonicity, answering a question posed by West.

Findings

 \, ext{alpha}(G) \, extless= 2 \, ext{kappa}(G)  guarantees prism-Hamiltonicity

The result strengthens the connection between independence number, connectivity, and prism-Hamiltonicity

Provides a sharp bound linking graph invariants to Hamiltonian properties of the prism

Abstract

The prism over a graph $G$ is the cartesian product $G \Box K_2$. It is known that the property of having a Hamiltonian prism (prism-Hamiltonicity) is stronger than that of having a $2$-walk (spanning closed walk using every vertex at most twice) and weaker than that of having a Hamilton path. For a graph $G$, it is known that $\alpha(G) \leq 2 \kappa(G)$, where $\alpha(G)$ is the independence number and $\kappa(G)$ is the connectivity, imples existence of a $2$-walk in $G$, and the bound is sharp. West asked for a bound on $\alpha (G)$ in terms of $\kappa (G)$ guaranteeing prism-Hamiltonicity. In this paper we answer this question and prove that $\alpha(G) \leq 2 \kappa(G)$ implies the stronger condition, prism-Hamiltonicity of $G$.