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Stanley-Reisner ideals of higher independence complexes of chordal graphs

Kanoy Kumar Das, Amit Roy, Kamalesh Saha

TL;DR

This work investigates the Stanley-Reisner ideals $J_t(G)$ of the $t$-independence complex $\mathrm{Ind}_t(G)$ for chordal graphs, linking algebraic invariants to the combinatorial structure of the induced hypergraph $\mathcal{H}(G,t)$. For every $t\ge 2$ and chordal $G$, it proves $\text{reg}(R/J_t(G))=(t-1)\nu_t(G)$ and $\text{pd}(R/J_t(G))=\text{bight}(J_t(G))$, where $\nu_t(G)$ is the induced matching number of $\mathcal{H}(G,t)$ and $\text{bight}$ is the big height. Consequences include a combinatorial characterization of when $J_t(G)$ has a linear resolution (iff $G$ is $t$-gap-free, i.e., $\nu_t(G)=1$) and a Cohen–Macaulay criterion (unmixedness), with both properties independent of the base field. These results generalize classical edge-ideal outcomes for chordal graphs and motivate higher-degree analogues (e.g., $t$-path and $t$-clique ideals).

Abstract

For $t\geq 2$, the $t$-independence complex $\mathrm{Ind}_t(G)$ of a graph $G$ is the collection of all $A\subseteq V(G)$ such that each connected component of the induced subgraph $G[A]$ has at most $t-1$ vertices. The topology of $\mathrm{Ind}_t(G)$ is intimately related to the combinatorial property of $G$. In this article, we consider the Stanley-Reisner ideal $J_{t}(G)$ of $\mathrm{Ind}_t(G)$ and focus on its algebraic properties. We prove that for a chordal graph $G$ and for all $t$ \[ \mathrm{reg}(R/J_{t}(G))=(t-1)ν_{t}(G) \text{ and } \mathrm{pd}(R/J_{t}(G))=\mathrm{bight}(J_{t}(G)), \] where $ν_{t}(G)$ denotes the induced matching number of the corresponding hypergraph of $J_{t}(G)$, and $\mathrm{reg}$, $\mathrm{pd}$ and $\mathrm{bight}$ stand for the regularity, projective dimension, and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley-Reisner ideal of the $t$-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs.

Stanley-Reisner ideals of higher independence complexes of chordal graphs

TL;DR

This work investigates the Stanley-Reisner ideals of the -independence complex for chordal graphs, linking algebraic invariants to the combinatorial structure of the induced hypergraph . For every and chordal , it proves and , where is the induced matching number of and is the big height. Consequences include a combinatorial characterization of when has a linear resolution (iff is -gap-free, i.e., ) and a Cohen–Macaulay criterion (unmixedness), with both properties independent of the base field. These results generalize classical edge-ideal outcomes for chordal graphs and motivate higher-degree analogues (e.g., -path and -clique ideals).

Abstract

For , the -independence complex of a graph is the collection of all such that each connected component of the induced subgraph has at most vertices. The topology of is intimately related to the combinatorial property of . In this article, we consider the Stanley-Reisner ideal of and focus on its algebraic properties. We prove that for a chordal graph and for all where denotes the induced matching number of the corresponding hypergraph of , and , and stand for the regularity, projective dimension, and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley-Reisner ideal of the -independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs.
Paper Structure (10 sections, 18 theorems, 48 equations, 2 figures)

This paper contains 10 sections, 18 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory and Combinatorics:
  4. Stanley-Reisner ideal of $\mathrm{Ind}_t(G)$:
  5. $J_t(G)$ as edge ideal of a hypergraph:
  6. Some algebraic invariants:
  7. Bounds on regularity and projective dimension:
  8. Castelnuovo-Mumford Regularity and Linearity
  9. Projective Dimension and Cohen-Macaulay Property
  10. Concluding Remarks

Key Result

Lemma 2.1

HaTrungTrung Let $I_1\subseteq R_1=\mathbb K[x_1,\ldots,x_n]$ and $I_2\subseteq R_2=\mathbb K[y_1,\ldots, y_m]$ be two graded ideals. Consider the ideal $I=I_1R+I_2R\subseteq R=\mathbb K[x_1,\ldots,x_n,y_1,\ldots,y_m]$. Then

Figures (2)

  • Figure 1: A chordal graph $G$.
  • Figure 2: $K_6$ and an edge removed.

Theorems & Definitions (37)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 27 more