The diagonal hypergraph of a matrix (bipartite graph)

Let A = [Uii] be a non-negative matrix. A diagonal of A is a set of positions D,,={(U(I). I). . . . . (O(H), II)) of A where u is a permutation of { !, . . . , n). A position of A is called positive if the entry of A from that position is positive. The diagonal D,, is called a positive diagonal provided all its positions are positive. Since the notion of a positive diagonal depends only on the zero-nonzero character of the entries of A, we assume henceforth that all matrices uve nzatrices of O’s a& 1’s. Alternatively we may say that the positivity of a diagonal of A depends only on the bipartite graph 1 ‘G(A) associated with A. This bipartite graph is defined by choosing two disjoi rt sets of n vertices X = {x,, . . . , x,,) and Y={y,,..., y,,} and putting an edge [Xi, yi] between Xi and yi if and only if aii # 0. The positive diagonals of A are then in one-to-one correspondence with the 1 -factors of BG(A), where a l-factor is a collection of n pairwise vertex disjoint edges. With the II x 11 matrix A we associate a hypergraph EM(A), called the diagoflal Itypergruph of A, as follows. The ver.ices of DH( 4) are the positive positions of A and the edges of DH(A) are the positive diagonals of A. (Alternatively we associate a hypergraph with a bipartite graph G whose vertices are the edges of G and whose edges are the I-factors of G.) If some positive position of A belongs to no positive diagonal of A, then the corresponding vertex of M-I(A) belongs to no edge, and we can safely delete that vertex without changing the important structure of DH(A). This is why wt. shall generally assume that every positive