The Pairing-Hamiltonian property in Cartesian products of graphs

Abstract

Let $G$ be a simple graph of even order at least four, and let $K_G$ denote the complete graph on $V(G)$. A perfect matching of $K_G$ is called a pairing of $G$. The graph $G$ has the Pairing-Hamiltonian property, or PH-property, if every pairing $M$ of $G$ admits a perfect matching $N\subseteq E(G)$, disjoint from $M$, such that $M\cup N$ is a Hamiltonian cycle of $K_G$. We prove that the PH-property is preserved under Cartesian products. More precisely, for graphs $G$ and $H$ of even order at least four, we show that both $G$ and $H$ are PH if and only if every pairing of $G\square H$ admits a Hamiltonian completion contained in a spanning union of vertex-disjoint prisms determined by a perfect matching of $G$ or of $H$. Without this support restriction, the converse fails: a Cartesian product may be PH even when neither of the two graphs is PH.