The Hirsch conjecture holds for normal flag complexes

TL;DR

This paper proves the Hirsch conjecture for all flag polytopes and flag homology manifolds by showing that flag and normal simplicial complexes satisfy the non-revisiting path conjecture, bounding their graph diameters.

Contribution

It establishes the Hirsch conjecture for flag polytopes and flag homology manifolds using a novel approach based on metric geometry and non-revisiting paths.

Findings

Holds for all flag polytopes

Validates for flag homology manifolds

Diameter bound matches the conjecture

Abstract

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes, and more generally, for all (connected) flag homology manifolds.