Georgia Institute of Technology College of Sciences

The problem of generating random integral tables from the set of all nonnegative integral tables with fixed marginals is of importance in statistics. The Diaconis-Sturmfels algorithm for this problem performs a random walk on the set of such tables. The moves in the walk are referred to as Markov bases and correspond to generators of a certain toric ideal. When only one and two-way marginals are considered, one can naturally associate a graph to the model. In this talk, I will present a characterization of all graphs for which the corresponding toric ideal can be generated in degree four, answering a question of Develin and Sullivant. I will also discuss some related open questions and demonstrate applications of the Four Color theorem and results on clean triangulations of surfaces, providing partial answers to these questions. Based on joint work with Daniel Kral and Ondrej Pangrac.