Characterize graphs whose edge ring K[G] is Cohen–Macaulay

Characterize all finite simple graphs G for which the standard graded edge ring K[G] = K[E]/IG is Cohen–Macaulay. Provide necessary and sufficient graph-theoretic conditions that determine when the depth of K[G] equals its Krull dimension.

Background

The paper studies how certain graph operations, especially path and edge contractions, affect the total Betti numbers and Cohen–Macaulayness of the edge ring K[G]. While many structural properties of toric ideals of graphs have been investigated, a complete graph-theoretic characterization of when K[G] is Cohen–Macaulay remains unknown.

This open problem is foundational: identifying the exact class of graphs whose edge rings are Cohen–Macaulay would unify and extend partial results and examples, and provide a clear combinatorial criterion linking graph structure to homological properties of the associated toric algebra.