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On Conjecture of Binomial Edge Ideals of Linear Type

Marie Amalore Nambi, Neeraj Kumar

TL;DR

This work addresses the conjecture of Jayanthan–Kumar–Sarkar that binomial edge ideals are of linear type for trees or unicyclic graphs. It introduces the p-sequence concept and shows that edge binomials of trees, and of a class of unicyclic graphs formed by adding pendant-vertex edges, can be organized into a p-sequence, enabling control of Rees-algebra relations and proving linear type for trees. A detailed description of the Rees-ideal in the tree case is provided, yielding explicit generators for the defining ideal of $\mathcal{R}(J_G)$. However, the paper also constructs a counterexample demonstrating that not all unicyclic graphs satisfy linear type, establishing a precise boundary for the conjecture and highlighting the need for structural conditions beyond being unicyclic.

Abstract

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$ is a tree or a unicyclic graph, then the binomial edge ideal of $G$ is of linear type. Our investigation validates this conjecture for trees. However, our study reveals that not all unicyclic graphs adhere to this conjecture.

On Conjecture of Binomial Edge Ideals of Linear Type

TL;DR

This work addresses the conjecture of Jayanthan–Kumar–Sarkar that binomial edge ideals are of linear type for trees or unicyclic graphs. It introduces the p-sequence concept and shows that edge binomials of trees, and of a class of unicyclic graphs formed by adding pendant-vertex edges, can be organized into a p-sequence, enabling control of Rees-algebra relations and proving linear type for trees. A detailed description of the Rees-ideal in the tree case is provided, yielding explicit generators for the defining ideal of . However, the paper also constructs a counterexample demonstrating that not all unicyclic graphs satisfy linear type, establishing a precise boundary for the conjecture and highlighting the need for structural conditions beyond being unicyclic.

Abstract

An ideal of a commutative ring is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if is a tree or a unicyclic graph, then the binomial edge ideal of is of linear type. Our investigation validates this conjecture for trees. However, our study reveals that not all unicyclic graphs adhere to this conjecture.
Paper Structure (10 sections, 11 theorems, 14 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 14 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Basic notions from graph theory.
  3. Basic results from commutative algebra.
  4. p-sequence
  5. Binomial edge ideals of p-sequence
  6. Ordering on edges of a tree
  7. Ordering on edges of a unicyclic graph
  8. On Conjecture
  9. Defining ideal of Rees algebra
  10. Counterexample

Key Result

Proposition 2.1

Let $A$ be a commutative ring. Suppose the sequence of elements $z_1,\ldots,z_n$ in $A$ forms a $p$-sequence. Then $z_{i+1},\ldots,z_n$ in $A/I_i$ forms a $p$-sequence, where $I_i=(z_1,\ldots,z_{i})$, for all $i$.

Figures (1)

  • Figure 1: The rooted tree $(G,1)$

Theorems & Definitions (42)

  • Conjecture A
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.1
  • Definition 1.2
  • Remark 1.5
  • Definition 1.3
  • Definition 1.4
  • ...and 32 more