Markov bases of binary graph models of K_4-minor free graphs
  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 $K_n$. 
Daniel Kr\'al', Serguei Norine, Ondrej Pangr\'ac