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

TL;DR

This paper provides a complete enumeration of the facets of cut polytopes for several highly symmetric graphs, confirming conjectures and solving problems relevant to quantum information theory.

Contribution

It offers the first complete facet lists for cut polytopes of specific symmetric graphs, confirming a conjecture about facet adjacency in $CUTP_8$ and addressing open problems in quantum information.

Findings

Complete facet lists for $K_8$, $K_{3,3,3}$, and other graphs.

Confirmation that all facets of $CUTP_8$ are adjacent to a triangle facet.

Solved problems in quantum information theory related to $K_{1,l,m}$ 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 $K_8$, $K_{3,3,3}$, $K_{1,4,4}$, $K_{5,5}$, some other $K_{l,m}$, $K_{1,l,m}$, $Prism_7, APrism_6$, M\"{o}bius ladder $M_{14}$, Dodecahedron, Heawood and Petersen graphs. For $K_8$, it shows that the huge lists of facets of the cut polytope $CUTP_8$ and cut cone $CUT_8$, given in [CR] is complete. We also confirm the conjecture that any facet of $CUTP_8$ is adjacent to a triangle facet. The lists of facets for $K_{1,l,m}$ with $(l,m)=(4,4),(3,5),(3,4)$ solve problems (see, for example, [Werner]) in quantum information theory.