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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.

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Correspondence to Jean Paul Zerafa.

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Communicated by Ararat Harutyunyan.

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Gauci, J.B., Zerafa, J.P. Perfect Matchings and Hamiltonicity in the Cartesian Product of Cycles. Ann. Comb. 25, 789–796 (2021). https://doi.org/10.1007/s00026-021-00548-1

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Keywords

  • Perfect matching
  • Hamiltonian cycle
  • Cartesian product of cycles

Mathematics Subject Classification

  • 05C70
  • 05C45
  • 05C76