The Pairing-Hamiltonian property in graph prisms

TL;DR

This paper extends the class of graphs known to have the PH-property by showing that prism graphs built from graphs with the PH-property also possess this property, including iterated prisms after a certain number of steps.

Contribution

It proves that the PH-property is preserved under the prism operation and establishes conditions for iterated prisms to have the PH-property.

Findings

Prism graphs of graphs with the PH-property also have the PH-property.

Existence of a threshold k₀ such that iterated prisms of a graph have the PH-property for all k ≥ k₀.

Abstract

Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$.