Diameter bound for facet-ridge incidence graphs of geometric lattices

TL;DR

This paper establishes an upper bound on the diameter of facet-ridge incidence graphs of geometric lattices' order complexes, using shelling techniques, and discusses the potential sharpness of this bound.

Contribution

It proves a new diameter bound for these graphs and explores the subtlety of whether this bound is tight, advancing understanding of geometric lattice structures.

Findings

Diameter of facet-ridge incidence graph is at most ${r race 2}$.

Shelling via atom orderings is key to the proof.

Evidence suggests the bound may be sharp, but the question remains subtle.

Abstract

This paper proves that the facet-ridge incidence graph of the order complex of any finite geometric lattice of rank $r$ has diameter at most ${r \choose 2}$. A key ingredient is the well-known fact that every ordering of the atoms of any finite geometric lattice gives rise to a lexicographic shelling of its order complex. The paper also gives results that provide some evidence that this bound ought to be sharp as well as examples indicating that the question of sharpness is quite subtle.