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Monomial ideals whose all matching powers are Cohen-Macaulay

Antonino Ficarra, Somayeh Moradi

TL;DR

This work addresses the problem of classifying monomial ideals whose matching powers are Cohen–Macaulay, focusing on edge ideals and the generalized notion of matching powers $I(G)^{[k]}$. It develops a hereditary framework showing CM for $I(G)^{[k]}$ passes to connected components and to subgraphs $G\setminus N_G[x]$, and provides a Tutte‑style combinatorial criterion for the last non‑vanishing power $I(G)^{[\nu(G)]}$. The authors obtain complete CM characterizations for graphs with a perfect matching under a dimension constraint, and extend the analysis to chordal and Cameron–Walker graphs, yielding explicit classifications. They complement the theory with computational results for small graphs (up to seven vertices), identifying exceptional cases and conjecturing field‑independence of the phenomenon.

Abstract

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analogue of the famous Tutte theorem regarding graphs having a perfect matching. For chordal graphs, very well-covered graphs and Cameron-Walker graphs, we completely solve our problem.

Monomial ideals whose all matching powers are Cohen-Macaulay

TL;DR

This work addresses the problem of classifying monomial ideals whose matching powers are Cohen–Macaulay, focusing on edge ideals and the generalized notion of matching powers . It develops a hereditary framework showing CM for passes to connected components and to subgraphs , and provides a Tutte‑style combinatorial criterion for the last non‑vanishing power . The authors obtain complete CM characterizations for graphs with a perfect matching under a dimension constraint, and extend the analysis to chordal and Cameron–Walker graphs, yielding explicit classifications. They complement the theory with computational results for small graphs (up to seven vertices), identifying exceptional cases and conjecturing field‑independence of the phenomenon.

Abstract

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analogue of the famous Tutte theorem regarding graphs having a perfect matching. For chordal graphs, very well-covered graphs and Cameron-Walker graphs, we completely solve our problem.
Paper Structure (4 sections, 22 theorems, 38 equations)

This paper contains 4 sections, 22 theorems, 38 equations.

Table of Contents

  1. Cohen-Macaulayness of matching powers of edge ideals
  2. Graphs with perfect matchings and very well-covered graphs
  3. Chordal graphs and Cameron-Walker graphs
  4. Edge ideals of small graphs whose all matching powers are Cohen-Macaulay

Key Result

Lemma 1.1

Let $I\subset S$ be a monomial ideal and $u\in S$ be a monomial with $u\notin I$. If $S/I$ is Cohen-Macaulay, then $S/(I:u)$ is Cohen-Macaulay and moreover $\textup{depth\,} S/I=\textup{depth\,} S/(I:u)$.

Theorems & Definitions (37)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • Lemma 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • Corollary 1.6
  • Example 1.7
  • ...and 27 more