Two Types of Matchings Extend to Hamiltonian Cycles in Hypercubes

Abstract

Ruskey and Savage posed the question: For \(n \geq 2\), does every matching in \(Q_n\) extend to a Hamiltonian cycle in \(Q_n\)? Fink showed that the answer is yes for every perfect matching, thereby proving Kreweras’ conjecture. In this paper, we prove that for \(n \geq 3\), every matching in \(Q_n\) not covering exactly two vertices at distance \(3\) extends to a Hamiltonian cycle in \(Q_n\). An edge in \(Q_n\) is an \(i\)-edge if its endpoints differ in the \(i\)th position. We also show that for \(n \geq 2\), every matching in \(Q_n\) consisting of edges in at most four types extends to a Hamiltonian cycle in \(Q_n\).