Bounds on the diameters of $r$-stacked and $k$-neighborly polytopes

TL;DR

This paper improves bounds on the diameters of certain polytopes and spheres, showing that for specific classes like $r$-stacked spheres with $r=O( ext{log} n)$, the polynomial Hirsch conjecture holds.

Contribution

It introduces a new bound on the diameter of normal simplicial complexes based on missing face size, leading to improved diameter bounds for $k$-neighborly spheres and $r$-stacked spheres.

Findings

New upper bounds on diameters of facet-ridge graphs

Polynomial Hirsch conjecture holds for $r$-stacked spheres with $r=O( ext{log} n)$

Enhanced understanding of diameter bounds in polytope theory

Abstract

We improve Larman's bound on the diameter of a polytope by showing that if $\Delta$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $\Delta$ is not larger than $2^{r-2}n$, where $n$ is the number of vertices of $\Delta$. We then use this result to provide new upper bounds on the diameters of the facet-ridge graphs of $k$-neighborly spheres, $r$-stacked spheres, and polytopes with small $g_r$. Specifically, our bounds imply that $r$-stacked spheres with $r=O(\log n)$ satisfy the polynomial Hirsch conjecture.