Polytopes with low excess degree

TL;DR

This paper investigates the structure and existence of $d$-polytopes with low excess degree, revealing new constraints on possible edge-vertex configurations and establishing structural properties for specific excess degrees.

Contribution

It provides new bounds on the excess degree of $d$-polytopes and characterizes their structure for certain excess degrees, extending previous results.

Findings

Excess degree cannot lie in [d+3, 2d-7] range.

Many pairs (f0, f1) are not realizable by any polytope.

Structural results for excess degrees d, d+2, and 2d-6.

Abstract

We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are not realised by any polytope. For $d$-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$.