Markov bases of binary graph models of K_4-minor free graphs

TL;DR

This paper characterizes graphs with Markov width at most four as exactly those without a $K_4$ minor, and establishes a lower bound on the Markov width for complete graphs, advancing understanding of graph invariants in algebraic statistics.

Contribution

It provides a complete characterization of graphs with bounded Markov width and introduces a lower bound for the Markov width of complete graphs.

Findings

Graphs with Markov width ≤ 4 are exactly those without a $K_4$ minor.

Markov width of $K_n$ grows at least as fast as $ ext{Omega}(n^{2- ext{epsilon}})$.

Answer to a question by Develin and Sullivant regarding Markov bases.

Abstract

Markov width of a graph is a graph invariant defined as the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We show that a graph has Markov width at most four if and only if it contains no $K_4$ as a minor, answering a question of Develin and Sullivant. We also present a lower bound of order $\Omega(n^{2-\varepsilon})$ on the Markov width of $K_n$.