Cohen-Macaulay $r$-partite graphs with minimal clique cover

TL;DR

This paper investigates conditions under which r-partite graphs have Cohen-Macaulay edge rings, proving the uniqueness of clique covers in certain cases, thus advancing understanding of algebraic properties of such graphs.

Contribution

It provides necessary conditions for r-partite graphs to have Cohen-Macaulay edge rings and establishes the uniqueness of clique covers in specific scenarios.

Findings

Necessary conditions for Cohen-Macaulay r-partite graphs.

Proof of clique cover uniqueness in certain Cohen-Macaulay graphs.

Insights into the algebraic structure of r-partite graphs.

Abstract

In this note, we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay. It is proved that if $G$ is an $r$-partite Cohen-Macaulay graph which is covered by some disjoint cliques of size $r$, then the clique cover is unique.