The Pairing-Hamiltonian property in graph prisms

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 (2007) [4] proved that for every $d\ge 2$, the d-dimensional hypercube $Q_d$ 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 $P\left(G\right)=G\square K_2$ 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 $k^{\text{th}}$-prism of a graph $P^k\left(G\right)$ has the PH-property for all $k\ge k_0$.