Doubly stochastic matrices and dicycle covers and packings in eulerian digraphs

Mirsky (1963) raised the question of characterizing Ω⁰_n, the convex hull of the nonidentity permutation matrices of order n, by a set of linear constraints. Cruse (1979) solved Mirsky's problem by presenting an implicit description of those constraints. We associate an eulerian digraph with each doubly stochastic matrix, and then restate Cruse's characterization of the polytope Ω⁰_n in terms of dicycle covers of these digraphs. Brualdi and Hwang (1992) have shown, by using Cruse's characterization and a result of Dridi (1980), an explicit set of linear inequalities that characterize Ω⁰_n for n ≤ 6. By using our characterization of Ω⁰_n, we show that their result is valid if and only if n ≤ 6. We show as well that if D is an eulerian digraph on n ≤ 6 nodes, then there is always a minimum dicycle cover which is integral. We apply this last result and a result of Seymour (1994) to derive a min-max relation for eulerian digraphs on n ≤ 6 nodes.