Cohen-Macaulay generalized binomial edge ideals

TL;DR

This paper classifies Cohen-Macaulay generalized binomial edge ideals associated with graphs, analyzing their properties such as unmixedness, especially focusing on bipartite and power cycle cases.

Contribution

It provides a complete classification of Cohen-Macaulay generalized binomial edge ideals and investigates their unmixedness in specific graph classes.

Findings

Classified Cohen-Macaulay generalized binomial edge ideals.

Identified conditions for unmixedness in bipartite graphs.

Analyzed unmixedness in power cycle graphs.

Abstract

Let $G$ be a simple graph on $n$ vertices and let $J_{G,m}$ be the generalized binomial edge ideal associated to $G$ in the polynomial ring $K[x_{ij}, 1\le i \le m, 1\le j \le n]$. We classify the Cohen-Macaulay generalized binomial edge ideals. Moreover we study the unmixedness and classify the bipartite and power cycle unmixed ones.