The cut cone III: On the role of triangle facets

The cut polytope P. is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph on n nodes. A well known class of facets of P. arises from the triangle inequalities: x o + Xlk + X~k < 2 and xlj -x~k -xjk < 0 for 1 < i, j, k < n. Hence, the metric polytope M., defined as the solution set of the triangle inequalities, is a relaxation of P.. We consider several properties of geometric type for P., in particular, concerning its position within M.. Strengthening the known fact ([3]) that P. has diameter 1, we show that any set of k cuts, k < log 2 n, satisfying some additional assumption, determines a simplicial face of iV/. and thus, also, of P.. In particular, the collection of low dimension faces of P. is contained in that of iV/,. Among a large subclass of the facets of P., the triangle facets are the closest ones to the barycentrum of P, and we conjecture that this result holds in general, The lattice generated by all even cuts (corresponding to bipartitions of the nodes into sets of even cardinality) is characterized and some additional questions on the links between general facets of P. and its triangle facets are mentioned.