Enumeration of the facets of cut polytopes over some highly symmetric graphs

Abstract

We report here a computation giving the complete list of facets for the cut polytopes over several very symmetric graphs with 15-30 edges, including K8, K3, 3, 3, K1, 4, 4, K5, 5, some other Kl,m, K1,l,m, Prism7,APrism6, Möbius ladder M14, dodecahedron, Heawood, and Petersen graphs. For K8, it shows that the huge list of facets of the cut polytope CUTP8 and cut cone CUT8, given in the literature is complete. We also confirm the conjecture that any facet of CUTP8 is adjacent to a triangle facet. The lists of facets for K1,l,m with (l,m)=(4,4),(3,5),(3,4) solve problems in the quantum information theory.