# Binomial edge ideals of bipartite graphs

## TL;DR

This paper classifies bipartite graphs whose binomial edge ideals are Cohen-Macaulay, characterizes their structure, and explores properties of related non-bipartite graphs, advancing understanding in algebraic combinatorics.

## Contribution

It provides a classification of bipartite graphs with Cohen-Macaulay binomial edge ideals and establishes a new equivalence involving the connectedness of dual graphs.

## Key findings

- Classified bipartite graphs with Cohen-Macaulay binomial edge ideals.
- Proved the converse of Hartshorne's result relating Cohen-Macaulayness and dual graph connectedness.
- Constructed bipartite graphs with unmixed but not Cohen-Macaulay ideals.

## Abstract

We classify the bipartite graphs $G$ whose binomial edge ideal $J_G$ is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. We study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with $J_G$ unmixed and not Cohen-Macaulay.