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On binomial edge ideals of corona of graphs

Buddhadev Hajra, Rajib Sarkar

Abstract

For a simple graph $G$, let $J_G$ denote the corresponding binomial edge ideal. This article considers the binomial edge ideal of the corona product of two connected graphs $G$ and $H$. The corona product of $G$ and $H$, denoted by $G\circ H$, is a construction where each vertex of $G$ is connected (via the coning-off) to an entire copy of $H$. This is a direct generalization of a cone construction. Previous studies have shown that for $J_{G \circ H}$ to be Cohen-Macaulay, both $G$ and $H$ must be complete graphs. However, there are no general formulae for the dimension, depth, or Castelnuovo-Mumford regularity of $J_{G\circ H}$ for all graphs $G$ and $H$. In this article, we provide a general formula for the dimension, depth and Castelnuovo-Mumford regularity of the binomial edge ideals of certain corona and corona-type (somewhat a generalization of corona) products of special interests. Additionally, we study the Cohen-Macaulayness, unmixedness and related properties of binomial edge ideals corresponding to above class of graphs. We have also added a short note on the reduction of the Bolognini-Macchia-Strazzanti Conjecture to all graphs with a diameter of $3$.

On binomial edge ideals of corona of graphs

Abstract

For a simple graph , let denote the corresponding binomial edge ideal. This article considers the binomial edge ideal of the corona product of two connected graphs and . The corona product of and , denoted by , is a construction where each vertex of is connected (via the coning-off) to an entire copy of . This is a direct generalization of a cone construction. Previous studies have shown that for to be Cohen-Macaulay, both and must be complete graphs. However, there are no general formulae for the dimension, depth, or Castelnuovo-Mumford regularity of for all graphs and . In this article, we provide a general formula for the dimension, depth and Castelnuovo-Mumford regularity of the binomial edge ideals of certain corona and corona-type (somewhat a generalization of corona) products of special interests. Additionally, we study the Cohen-Macaulayness, unmixedness and related properties of binomial edge ideals corresponding to above class of graphs. We have also added a short note on the reduction of the Bolognini-Macchia-Strazzanti Conjecture to all graphs with a diameter of .

Paper Structure

This paper contains 11 sections, 28 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure and main results of the paper
  3. Preliminaries
  4. Combinatorial Preliminaries
  5. Algebraic Preliminaries
  6. Unmixedness, Cohen-Macaulayness, and Accessibility of Certain Corona Products
  7. A small remark on a conjecture of Bolognini-Macchia-Strazzanti
  8. Dimension, Depth and Regularity of Certain Corona Products
  9. Dimension of corona with complete graph
  10. Depth and regularity of corona with complete graph
  11. Depth and regularity of corona with Cohen-Macaulay closed graph

Key Result

Theorem A

Let $G$ and $H$ be two graphs such that $|V(G)|>1$. For a non-empty subset $L$ of $V(G)$, if $J_{G\circ_L H}$ is unmixed (resp. accessible), then $J_H$ is also unmixed (resp. accessible).

Figures (1)

  • Figure 1: The corona product

Theorems & Definitions (52)

  • Theorem A: Theorems \ref{['Thm: Unmixedness of L-corona of G and H implies H is unmixed']}, \ref{['Thm: Acc. system of L-corona of G and H implies H has acc. system']}
  • Theorem B: Theorem \ref{['Thm: Combined -- l-corona with complete is unmixed, accessible, CM iff the other is unmixed, accessible, CM respectively']}
  • Theorem C: Theorem \ref{['Thm: Dimension of l-corona with complete graph']}
  • Theorem D: Theorem \ref{['Thm: Depth for corona with complete graph']}
  • Theorem E: Theorem \ref{['Thm: Depth for corona with CM closed graph']}
  • Theorem 2.1: Auslander--Buchsbaum Formula
  • Lemma 2.2: Ohtani, cf. Oht11
  • Lemma 2.3: Depth Lemma
  • Lemma 2.4: Regularity Lemma
  • Proposition 3.1
  • ...and 42 more