Perfect Matchings and Hamiltonicity in the Cartesian Product of Cycles

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