In the last decades the connections between Commutative Algebra and Combinatorics have been extensively explored. In this perspec-
tive, many authors have considered classes of ideals in a polynomial ring that can be naturally associated with combinatorial objects, and
have studied their algebraic invariants exploiting this combinatorial connection.
In this talk we are interested in the so-called binomial edge ideals, which are ideals generated by binomials corresponding to the edges of
a finite simple graph. They can be viewed as a generalization of the ideal of the maximal minors of a generic matrix with two rows, where
only some minors are considered.
After reviewing some results about these ideals, we present a conjecture for a combinatorial characterization of Cohen-Macaulay binomial
edge ideals. We identify sucient and necessary conditions for Cohen-Macaulayness, both of which can be read o from the underlying graph.
Moreover, we show that these conditions are indeed equivalent for large classes of graphs settling the conjecture in these cases.
This is joint work with Davide Bolognini and Antonio Macchia.