Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension

TL;DR

This paper classifies sequentially Cohen-Macaulay co-chordal graphs using ($d_1$,$d_2$,...,$d_q$)-trees, explores their projective dimension, and establishes bounds related to maximum vertex degree.

Contribution

It introduces a complete classification of sequentially Cohen-Macaulay co-chordal graphs via ($d_1$,$d_2$,...,$d_q$)-trees and analyzes the projective dimension in relation to vertex degree.

Findings

Complete classification of sequentially Cohen-Macaulay co-chordal graphs.

The projective dimension is at least the maximum vertex degree.

Equality in projective dimension and maximum vertex degree occurs under specific conditions.

Abstract

We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially Cohen-Macaulay co-chordal graphs. This class also yields a classification of bi-sequentially Cohen-Macaulay graphs. We study the relationship between the projective dimension of a graph and its maximum vertex degree. We show that the projective dimension is always at least the maximum vertex degree, although this bound is not always tight, even for co-chordal graphs. However, equality holds when the graph is sequentially Cohen-Macaulay co-chordal or has a full vertex.