Higher connectivity of fiber graphs of Gr\"obner bases

TL;DR

This paper investigates the connectivity properties of fiber graphs derived from Gr"obner bases of contingency tables, providing combinatorial results that support a conjecture linking connectivity to minimum vertex degree, with implications for statistical algorithms.

Contribution

It offers new combinatorial insights into fiber graph connectivity, confirming a conjecture relating connectivity to minimum vertex degree in this context.

Findings

Connectivity equals minimum vertex degree in studied fiber graphs

Supports Engstr"om's conjecture on fiber graph connectivity

Uses elementary combinatorial techniques for analysis

Abstract

Fiber graphs of Gr\"obner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these graphs using the Metropolis-Hastings algorithm. The connectivity of the graphs has implications on how fast the algorithm converges. In this paper, we study a class of fiber graphs with elementary combinatorial techniques and provide results that support a recent conjecture of Engstr\"om: the connectivity is given by the minimum vertex degree.