Competition periods of multipartite tournaments

The adjacency matrix of a k-partite tournament for k >= 2 is represented as a (0, 1) Boolean block matrix A with blocks A(ij) for 1 <= i, j <= k such that A(ii) is a zero matrix; A(ij) + A(ji)(T) is a matrix with all elements equal to 1 for i not equal j where A(ji)(T) means the transpose of A(ji). There is the smallest positive integer q such that A(q+i)(A(T))(q+i) = A(q+r+i)(A(T))(q+r+i) for some positive integer r and every nonnegative integer i. Then there is also the smallest positive integer p such that A(q)(A(T))(q) = A(q+p)(A(T))(q+p). In this paper, we show that p <= 3, and, further, if k = 2, then p <= 2.