Integral matrices with given row and column sums

Abstract

Let P = ( p ij ) and Q = ( q ij ) be m × n integral matrices, R and S be integral vectors. Let U P Q ( R, S) denote the class of all m × n integral matrices A with row sum vector R and column sum vector S satisfying P ⩽ A ⩽ Q. For a wide variety of classes U P Q ( R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in U P Q ( R, S). The first characterization unifies the results of Gale-Ryser, Fulkerson, and Anstee. Many other properties of (0, 1)-matrices with prescribed row and column sum vectors generalize to integral classes satisfying the main condition. We also study the decomposibility of integral classes satisfying the main condition. As a consequence of our decomposibility theorem, it follows a theorem of Beineke and Harary on the existence of a strongly connected digraph with given indegree and outdegree sequences. Finally, we introduce the incidence graph for a matrix in an integral class U P Q ( R, S) and study the invariance of an element in a matrix in terms of its incidence graph. Analogous to Minty's Lemma for arc colorings of a digraph, we give a very simple labeling algorithm to determine if an element in a matrix is invariant. By observing the relationship between invariant positions of a matrix and the strong connectedness of its incidence graph, we present a very short graph theoretic proof of a theorem of Brualdi and Ross on invariant sets of (0, 1)-matrices. Our proof also implies an analogous theorem for a class of tournament matrices with given row sum vector, as conjectured by the analogy between bipartite tournaments and ordinary tournaments.