Abstract In this paper, we investigate the maximal degree of minimal generators of the toric ideal of the matching polytope of a graph. It is known that the toric ideal associated with a bipartite graph is generated by binomials of degree at most 3. We show that this fact is equivalent to a result in the theory of edge colorings of bipartite multigraphs. Moreover, a characterization of bipartite graphs whose toric ideals are generated by quadratic binomials is given. Finally, we discuss the maximal degree of minimal generators of the toric ideal associated with a general graph and give a conjecture.
Mori et al. (Wed,) studied this question.