Matching graphs of Hypercubes and Complete Bipartite Graphs

Kreweras’ conjecture [1] asserts that every perfect matching of the hypercube Qd can be extended to a Hamiltonian cycle. We [2] proved this conjecture but here we present a simplified proof. The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph M(Qd) of the d-dimensional hypercube is bipartite for d ≥ 2 and connected for d ≥ 4. This proves another Kreweras ’ conjecture [1] that the graph Md is connected, where Md is obtained from M(Qd) by contracting every pair of vertices of M(Qd) whose corresponding perfect matchings are isomorphic.