Summary: The 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 $\varOmega (n^{2 - ε})$ on the Markov width of $K_n$.