The Maximum Number of Disjoint Permutations Contained in a Matrix of Zeros and Ones

Abstract

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.