Combinatorics of (parity) binomial edge ideals

A binomial ideal is an ideal of a polynomial ring generated by binomials. It appears in various areas of commutative algebra and combinatorics as well as in statistics. Recently, one is especially interested in two classes of binomial ideals arising from graphs: the binomial edge ideal and the parity binomial edge ideal. A polular theme is how the combinatorial properties of a graph are encoded in the Cohen-Macaulayness of these ideals. In the case of bipartite graph, this problem is solved in work of Bolognini, Macchia, and Strazzant. The goal of this project is to study the problem for non-bipartite graphs.