Cohen-Macaulay Circulant Graphs

TL;DR

This paper investigates conditions under which circulant graphs are Cohen-Macaulay, focusing on well-covered graphs and characterizing Cohen-Macaulay cubic circulant graphs, revealing that Cohen-Macaulay property is not preserved under lexicographical products.

Contribution

It characterizes Cohen-Macaulay circulant graphs, especially well-covered and cubic cases, and shows the Cohen-Macaulay property is not preserved under lexicographical products.

Findings

Cohen-Macaulay circulant graphs are characterized among well-covered families.

Cubic circulant graphs that are Cohen-Macaulay are identified.

Cohen-Macaulay property is not preserved under lexicographical graph products.

Abstract

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.