Regularity and projective dimension of edge ideal of $C_5$-free vertex decomposable graphs

TL;DR

This paper investigates algebraic invariants like regularity, projective dimension, and depth of edge ideals for specific graph classes, especially $C_5$-free vertex decomposable graphs, linking them to graph invariants.

Contribution

It provides explicit formulas and characterizations for regularity, projective dimension, and depth of edge ideals in $C_5$-free vertex decomposable graphs, extending to forests and Cohen-Macaulay bipartite graphs.

Findings

Regularity equals the maximum number of 3-disjoint edges in the graph.

Characterization of projective dimension and depth for the class of graphs.

Descriptions of invariants for forests and Cohen-Macaulay bipartite graphs.

Abstract

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of 3-disjoint edges in $G$. Moreover for this class of graphs we characterize $\T{pd}(R/I(G))$ and $\T{depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.