Circulant $S_2$ graphs

TL;DR

This paper characterizes when circulant graphs satisfy Serre's condition S_2, linking it to properties like well-coveredness, Buchsbaumness, and Cohen-Macaulayness, and provides examples of graphs with specific property combinations.

Contribution

It offers a complete characterization of circulant graphs satisfying S_2, connecting this condition with existing graph properties and providing new examples.

Findings

S_2 condition is equivalent to well-coveredness or Buchsbaumness for some families.

S_2 condition is equivalent to Cohen-Macaulayness for other families.

Examples of infinite circulant graph families are provided with specific property distinctions.

Abstract

Recently, Earl, Vander Meulen, and Van Tuyl characterized some families of Cohen-Macaulay or Buchsbaum circulant graphs discovered by Boros-Gurvich-Milani$\check{\text{c}}$, Brown-Hoshino, and Moussi. In this paper, we will characterize those families of circulant graphs which satisfy Serre's condition $S_2$. More precisely, we show that for some families of circulant graphs, $S_2$ property is equivalent to well-coveredness or Buchsbaumness, and for some other families it is equivalent to Cohen-Macaulayness. We also give examples of infinite families of circulant graphs which are Buchsbaum but not $S_2$, and vice versa.