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

Chwas Ahmed, Amir Mafi, Mohammed Rafiq Namiq

TL;DR

This work classifies sequentially Cohen–Macaulay co-chordal graphs by introducing the $(d_1,...,d_q)$-tree family and proving a graph G is sequentially Cohen–Macaulay in the co-chordal setting iff its complement is such a tree. It further clarifies the relationship between projective dimension and maximum vertex degree, proving a universal lower bound and identifying precise equality conditions for two broad graph classes: those with a full vertex and those whose complement is a $(d_1,...,d_q)$-tree. The results connect the combinatorial structure of graph complements with algebraic properties of edge ideals and Stanley–Reisner rings, refining conjectures and extending prior work on the pd–degree relationship. Together, they provide practical criteria for recognizing sequentially Cohen–Macaulay co-chordal graphs and deepen understanding of how graph topology governs projective-dimension phenomena.

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.

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

TL;DR

This work classifies sequentially Cohen–Macaulay co-chordal graphs by introducing the -tree family and proving a graph G is sequentially Cohen–Macaulay in the co-chordal setting iff its complement is such a tree. It further clarifies the relationship between projective dimension and maximum vertex degree, proving a universal lower bound and identifying precise equality conditions for two broad graph classes: those with a full vertex and those whose complement is a -tree. The results connect the combinatorial structure of graph complements with algebraic properties of edge ideals and Stanley–Reisner rings, refining conjectures and extending prior work on the pd–degree relationship. Together, they provide practical criteria for recognizing sequentially Cohen–Macaulay co-chordal graphs and deepen understanding of how graph topology governs projective-dimension phenomena.

Abstract

We introduce a class of chordal graphs called (,,,)-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.
Paper Structure (5 sections, 12 theorems, 17 equations, 9 figures)

This paper contains 5 sections, 12 theorems, 17 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $(d_1,d_2,\dots,d_q)$-tree graphs
  4. Relation between Projective dimension of a graph and its maximum degrees
  5. The projective dimension and maximum degree is equal for certain of classes of graphs

Key Result

Theorem 3.2

Let $G$ be a graph. Then $\overline{G}$ is a $(d_1,d_2,\dots,d_q)$-tree if and only if $\Delta_G=\left\langle F_1,F_2,\dots,F_q\right\rangle$ is a vertex decomposable (hence shellable and sequentially Cohen--Macaulay) quasi-forest simplicial complex.

Figures (9)

  • Figure 1: Examples of generalised $d$-tree graphs
  • Figure 2: Two non-isomorphic $(3,3,2)$-tree graphs.
  • Figure 5: A visual representation of the graph $G$ and the intermediate graphs $G_1$ and $G_2$ obtained during the process
  • Figure 6: The Graph $G$ and Intermediate Graphs $G_1$ and $G_2$ during the Process
  • Figure 7: The $4$-barbell graph
  • ...and 4 more figures

Theorems & Definitions (38)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Example 3.7
  • Lemma 4.1
  • ...and 28 more