Transformation from arbitrary matchings to stable matchings

D. E. Knuth (1976, "Mariages stables," Presses Univ. Montréal, Montreal) conjectured that any matching can be transformed to some stable matching by a sequence of b-interchanges. Given a matching M and a blocking pair (m, w) for M, a b-interchange for M by (m, w) is defined as a transformation from M to a matching obtained by replacing two pairs (m, p_M (m)) and (p_M (w), w) in M with (m, w) and (p_M (w), p_M (m)). In this paper, we give a counter-example in which some matching cannot be transformed to any stable matching by b-interchanges. However, any matching can be transformed to some stable matching by using b-interchanges and identifying special cycling. We also give an algorithm to find either such cycling or a stable matching.