A complete characterization of $(f_0, f_1)$-pairs of 6-polytopes

TL;DR

This paper provides a complete characterization of the pairs of the number of vertices and edges for 6-dimensional polytopes, extends results to certain 7-dimensional cases, and proposes conjectured bounds for higher dimensions.

Contribution

It offers a complete description of (f0, f1)-vector pairs for 6-polytopes and extends partial results to 7-polytopes with high excess degree, proposing new conjectures for general dimensions.

Findings

Complete characterization for 6-polytopes' (f0, f1) pairs

Characterization for 7-polytopes with excess degree > 11

Conjectured bounds for (f0, f1) pairs in higher dimensions

Abstract

We completely characterize the first two entries, namely the $(f_0, f_1)$-vector pairs, for $6$-dimension polytopes. We also find the characterization for $7$-dimension polytopes with excess degree greater than $11$ and, we conjecture bounds fulfilled by $(f_0, f_1)$-vector pairs for any $d$-polytope having an excess degree greater than $3d-10$.