Characterization of the types of maximum noninteger vertices in the relaxation polyhedron of the four-index axial assignment problem

For the relaxation polyhedron $M(4,n)$ in the four-index axial assignment problem of order $n$, $n\geqslant 3$, a characterization of all possible types (except for a single case) of maximum noninteger vertices, i.e., vertices with $4n-3$ fractional components is proposed. A formula enumerating all the maximum noninteger vertices of the same type in $M(4,n)$ is derived.