All 2-neighborly d-polytopes with at most d + 9 facets

TL;DR

This paper provides a complete enumeration of 2-neighborly d-polytopes with up to d+9 facets, revealing new bounds and specific realizations, including exceptions outside 0/1-polytope constructions.

Contribution

It offers a comprehensive classification of 2-neighborly d-polytopes with up to d+9 facets, including the identification of non-0/1-polytope examples and updated facet-vertex bounds.

Findings

All such polytopes are 0/1-polytopes except one 6-polytope and its pyramids.

The number of facets is at least the number of vertices for polytopes with up to d+10 facets.

Updated lower bounds for the facets of 2-neighborly d-polytopes.

Abstract

We give a complete enumeration of all 2-neighborly $d$-polytopes with $d+9$ and less facets. All of them are realized as 0/1-polytopes, except a 6-polytope $P_{6,10,15}$ with 10 vertices and 15 facets, and pyramids over $P_{6,10,15}$. In particular, we update the lower bounds for the number of facets of a 2-neighborly $d$-polytope $P$ and showed that the number of facets of $P$ is not less than the number of its vertices $f_0(P)$ for $f_0(P) \le d + 10$.