Perfect Matchings and Hamiltonicity in the Cartesian Product of Cycles

Abstract

If every perfect matching of a graph G extends to a Hamiltonian cycle, we shall say that G has the PMH-property—a concept first studied in the 1970s by Las Vergnas and Häggkvist. A pairing of a graph G is a perfect matching of the complete graph having the same vertex set as G. A somewhat stronger property than the PMH-property is the following. A graph G has the PH-property if every pairing of G can be extended to a Hamiltonian cycle of the underlying complete graph using only edges from G. The name for the latter property was coined in 2015 by Alahmadi et al.; however, this was not the first time this property was studied. In 2007, Fink proved that every n-dimensional hypercube, for $n\ge 2$, has the PH-property. After characterising all the cubic graphs having the PH-property, Alahmadi et al. attempt to characterise all 4-regular graphs having the same property by posing the following problem: for which values of p and q does the Cartesian product $C_p\square C_q$ of two cycles on p and q vertices have the PH-property? We here show that this only happens when both p and q are equal to four, namely for $C_{4}\square C_{4}$, the 4-dimensional hypercube. For all other values, we show that $C_{p}\square C_{q}$ does not even admit the PMH-property.