Prime interchange graphs of classes of matrices of zeros and ones

Abstract

Let R = ( r 1,…, r m ) and S = ( s 1,…, s n ) be nonnegative integral vectors, and let U ( R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of U ( R, S) is a position whose entry is the same for all matrices in U ( R, S). The interchange graph G( R, S) is the graph where the vertices are the matrices in U ( R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r i ≤ n − 1 ( i = 1,…, m) and 1 ≤ s j ≤ m − 1 ( j = 1,…, n), G( R, S) is prime if and only if U ( R, S) has no invariant positions.