Generalized transitive tournaments and doubly stochastic matrices

Abstract We show that a doubly stochastic matrix is a convex combination of nonidentity permutation matrices if and only if it can be written as the sum of a nonnegative matrix and a convex combination of cycle matrices. We use this result to give a shorter proof of the theorem of Cruse which asserts that a doubly stochastic matrix is a convex combination of nonidentity permutation matrices if and only if its inner product with each generalized transitive tournament matrix is at least 1. The generalized transitive tournaments of order n form a convex polytope T n which contains the convex hull T n * (also called the linear ordering polytope) of the transitive tournaments. Each transitive tournament matrix of order n is an extreme point of T n , but for n ⩾ 6 there are other extreme points. With each generalized tournament matrix T of order n we associate a graph whose edges correspond to the nonintegral entries of T . We investigate which graphs can arise from generalized transitive tournaments and which can arise from extreme generalized transitive tournaments. We briefly discuss a generalization of the linear ordering polytope to arbitrary posets.