Abstract :
The metric polytope MPn is defined by the triangle inequalities: xij − xik − xjk ⩽ 0 and xij + xik + xjk ⩽ 2 for all triples i, j, k of (1, …, n). The integral vertices of MPn are the incidence vectors of the cuts of the complete graph Kn. Therefore, MPn is a relaxation of the cut polytope of Kn. We study here the fractional vertices of MPn. Many of them are constructed from graphs; this is the case for the one-third-integral vertices. One-third-integral vertices are, in a sense, the simplest fractional vertices of MPn as MPn has no half-integral vertices. Several constructions for one-third-integral vertices are presented. In particular, the graphic vertices arising from the suspension of a tree are characterized. We describe the symmetries of MPn, and obtain that the vertices are partitioned into switching classes. With the exception of the cuts which are pairwise adjacent, it is shown that no two vertices of the same switching class are adjacent on MPn. The question of adjacency of the fractional vertices to the integral ones is also addressed. All the vertices of MPn for n ⩽ 6 are described.
