Matching graphs of Hypercubes and Complete Bipartite Graphs

Krewerasʹ conjecture [G. Kreweras: Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996) 87–91] asserts that every perfect matching of the hypercube Q d can be extended to a Hamiltonian cycle. We [J. Fink: Perfect Matchings Extend to Hamilton Cycles in Hypercubes, to appear in J. Comb. Theory, Series B] proved this conjecture but here we present a simplified proof. tching 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 ( Q d ) of the d-dimensional hypercube is bipartite for d ≥ 2 and connected for d ≥ 4 . This proves another Krewerasʹ conjecture [G. Kreweras: Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996) 87–91] that the graph M d is connected, where M d is obtained from M ( Q d ) by contracting every pair of vertices of M ( Q d ) whose corresponding perfect matchings are isomorphic.