Combinatorial properties of noninteger vertices of a polytope in a three-index axial assignment problem

It is proved that, for any r ? { 2n, 2n + 1,?, 3n?2} and only for such r, the polytope of a three-index axial assignment problem of order n, n ? 2, contains completely r-noninteger vertices (r-CNVs), i.e., vertices such that all their positive components are fractional and their number equals r. For each r ? {2n, 2n + 1,?, 3n ?2}, all the types of r-CNVs are characterized and the combinatorial properties of completely r-noninteger vertices of the polytope are studied.