Matching graphs of hypercubes and complete bipartite graphs

Abstract

Kreweras’ 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 of Q d . We [Jiří Fink, Perfect matchings extend to hamilton cycles in hypercubes, J. Combin. Theory Ser. B, 97 (6) (2007) 1074–1076] 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 of G . We show that the matching graph M ( K n , n ) of a complete bipartite graph is bipartite if and only if n is even or n = 1 . We prove that M ( K n , n ) is connected for n even and M ( K n , n ) has two components for n odd, n ≥ 3 . We also compute distances between perfect matchings in M ( K n , n ) .