Full classification of Cohen–Macaulay binomial edge ideals

Establish a complete classification of simple graphs G for which the binomial edge ideal J_G in the standard graded polynomial ring S = K[x_1,...,x_n,y_1,...,y_n] is Cohen–Macaulay.

Background

The paper studies sequentially Cohen–Macaulay binomial edge ideals and surveys known results on Cohen–Macaulayness. While numerous advances have been made, a definitive characterization of all graphs yielding Cohen–Macaulay binomial edge ideals has not been achieved.

This open problem is foundational in the area: binomial edge ideals connect graph combinatorics with commutative algebra, and Cohen–Macaulayness is a central property. A full classification would unify and extend several partial results across various graph families.