An improved upper bound on the diameters of subset partition graphs

TL;DR

This paper establishes a new subexponential upper bound on the diameters of subset partition graphs, extending previous bounds for polyhedra and connected layer families, thus advancing understanding of combinatorial properties of these graphs.

Contribution

It introduces an improved upper bound on subset partition graph diameters, analogous to recent bounds for polyhedra, based on properties related to connectivity.

Findings

Proves a new upper bound on subset partition graph diameters.

Extends bounds known for polyhedra to a broader class of graphs.

Provides theoretical insights into the structure of subset partition graphs.

Abstract

In 1992, Kalai and Kleitman proved the first subexponential upper bound for the diameters of convex polyhedra. Eisenbrand et al. proved this bound holds for connected layer families, a novel approach to analyzing polytope diameters. Very recently, Todd improved the Kalai-Kleitman bound for polyhedra to $(n-d)^{1+\log_2d}$. In this note, we prove an analogous upper bound on the diameters of subset partition graphs satisfying a property related to the connectivity property of connected layer families.