Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r1,…, rm) and S = (s1,…, sn) 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 ≤ ri ≤ n − 1 (i = 1,…, m) and 1 ≤ sj ≤ m − 1 (j = 1,…, n), G(R, S) is prime if and only if U(R, S) has no invariant positions.