On joint realization of (0, 1) matrices

Abstract

Let R=( r 1, r 2,…, r m ), S=( s 1, s 2,…, s n ), R′=( r′ 1, r′ 2,… , r′ m ), and S′= ( s′ 1, s′ 2, s′ n be nonnegative integral vectors. Denote by A(R,S) the class of (0,1) matrices with row sum vector R and column sum vector S. The three classes A(R,S), A(R′,S′) , and A(R+R′,S+S′) are called jointly realizable if there exist a matrix A in A(R,S) and a matrix B in A(R′,S′) such that A+Bϵ A(R+R′,S+S′) . In this paper, we prove that if A(R,S), A(R′,S′) , and A(R+R′,S+S′) are nonempty but not jointly realizable, then in the first two classes there must exist a matrix having one of the following unavoidable configurations: ▪, ▪. A similar theorem is proved about unavoidable configurations in A(R+R′,S+S′) . We also give a slight generalization of a theorem of Anstee, regarding joint realization of matrices where one of the classes has row sums differing by at most 1, along with a very short proof.