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Squarefree powers of closed neighborhood ideals

Marie Amalore Nambi, Ayesha Asloob Qureshi

Abstract

In this article, we characterize all trees whose highest non-vanishing squarefree power of the closed neighborhood ideal is componentwise linear. In addition, we investigate the Castelnuovo-Mumford regularity of the $ν$-th squarefree power of the closed neighborhood ideal of trees and show that this number can be arbitrarily larger than the degree of the ideal. Finally, we give a formula for the regularity of $ν$-th squarefree power of the closed neighborhood ideal of caterpillar graphs.

Squarefree powers of closed neighborhood ideals

Abstract

In this article, we characterize all trees whose highest non-vanishing squarefree power of the closed neighborhood ideal is componentwise linear. In addition, we investigate the Castelnuovo-Mumford regularity of the -th squarefree power of the closed neighborhood ideal of trees and show that this number can be arbitrarily larger than the degree of the ideal. Finally, we give a formula for the regularity of -th squarefree power of the closed neighborhood ideal of caterpillar graphs.
Paper Structure (6 sections, 22 theorems, 69 equations, 6 figures)

This paper contains 6 sections, 22 theorems, 69 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Required Ingredients
  3. Componentwise Linearity and Linear Relations
  4. Some lemmas on matching of simplicial trees
  5. Componentwise linearity of highest non-vanishing squarefree power for trees
  6. Regularity of the highest squarefree power

Key Result

Lemma 2.5

HT2010 Let $I$ be a homogeneous ideal of $S$ and $f$ be a degree $d$ monomial of $S$. Consider $0 \rightarrow S/(I:f) (-d)\rightarrow S/(I) \rightarrow S/(I+f) \rightarrow 0$ the short exact sequence. Then one has The equality holds if $\mathop{\mathrm{reg}}\nolimits(S/(I:f))+d \neq \mathop{\mathrm{reg}}\nolimits(S/I)$.

Figures (6)

  • Figure 1: The tree $\mathcal{T}$.
  • Figure 2: The neighborhoods of the red vertices belong to $V_U$, while the neighborhoods of the blue vertices belong to $V_N$.
  • Figure 3: The neighborhoods of the red vertices belong to $V_U$, while the neighborhoods of the blue vertices belong to $V_N$.
  • Figure 4: The neighborhoods of the red vertices belong to $V_U$ and the the neighborhoods of the blue vertices belong to $V_N$.
  • Figure 5: The graph $G_1$.
  • ...and 1 more figures

Theorems & Definitions (52)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • Example 2.6
  • Theorem 2.7
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 42 more