On minimally one-factorable r-regular bipartite graphs

A corollary to the famous Marriage Theorem by Ph. Hall [7] says that every r-regular bipartite graph is one-factorable (cf. e.g. [10, Theorem 3:2]); a glance at the proof reveals that this theorem admits the following slightly sharper formulation: let G be an r-regular bipartite graph; then G has one-factors and every one-factor can be completed to a one-factorization. We will prove that the Heawood graph is an instance of a graph ful lling this theorem minimally, i.e. in which every one-factor belongs to precisely one one-factorization. When dealing with r-regular bipartite graphs G and their adjacency matrices A, useful tools are the following algebraic invariants: