The edge rings of compact graphs

TL;DR

This paper classifies compact graphs, characterized by no even cycles and satisfying the odd-cycle condition, and analyzes their edge rings, revealing relationships between algebraic invariants and graph properties.

Contribution

It provides a complete classification of compact graphs and establishes explicit formulas for the Cohen-Macaulay type, projective dimension, and regularity of their edge rings.

Findings

Cohen-Macaulay type equals the number of induced cycles minus one.

Projective dimension equals the number of induced cycles minus one.

Regularity equals the matching number of the graph after removing degree-one vertices.

Abstract

We define a simple graph as compact if it lacks even cycles and satisfies the odd-cycle condition. Our focus is on classifying all compact graphs and examining the characteristics of their edge rings. Let $G$ be a compact graph and $\mathbb{K}[G]$ be its edge ring. Specifically, we demonstrate that the Cohen-Macaulay type and the projective dimension of $\mathbb{K}[G]$ are both equal to the number of induced cycles of $G$ minus one, and that the regularity of $\mathbb{K}[G]$ is equal to the matching number of $G_0$. Here, $G_0$ is obtained from $G$ by removing the vertices of degree one successively, resulting in a graph where every vertex has a degree greater than 1.