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Linear strands of powers of certain binomial edge ideals

Abbas Dohadwala, Bryan Flores-Silva, Alicia Orozco-Moya, Zoe Siegelnickel

TL;DR

The paper analyzes linear-strand Betti numbers of all powers of binomial edge ideals $J_G$ for closed graphs $G$, focusing on the $K_4$-free case. Using Rees-algebra techniques and the structure of EN-type and Koszul-type relations (with no Plücker relations when $G$ is $K_4$-free), it derives a closed formula for the linear strand: $$\beta_{i,2m+i}(J_G^m)=\binom{e+m-i-1}{m-i}\binom{2t}{i}$$, where $e$ counts edges and $t$ counts triangles. The result shows that these Betti numbers coincide with those of the lex initial ideal $\mathrm{in}_{lex} J_G^m$, confirming the Ene–Rinaldo–Terai conjecture in this special case. This provides explicit, combinatorial control of the linear strands in powers and advances understanding of the interaction between binomial edge ideals, their initial ideals, and Rees-algebra syzygies.

Abstract

We provide a closed formula for the graded Betti numbers in the linear strands of all powers of binomial edge ideals $J_G$ arising from closed graphs $G$ that do not have the complete graph $K_4$ as an induced subgraph. We show that these agree with the corresponding Betti numbers for the powers of the lexicographic initial ideal of $J_G$, thereby confirming a conjecture of Ene--Rinaldo--Terai in a special case.

Linear strands of powers of certain binomial edge ideals

TL;DR

The paper analyzes linear-strand Betti numbers of all powers of binomial edge ideals for closed graphs , focusing on the -free case. Using Rees-algebra techniques and the structure of EN-type and Koszul-type relations (with no Plücker relations when is -free), it derives a closed formula for the linear strand: , where counts edges and counts triangles. The result shows that these Betti numbers coincide with those of the lex initial ideal , confirming the Ene–Rinaldo–Terai conjecture in this special case. This provides explicit, combinatorial control of the linear strands in powers and advances understanding of the interaction between binomial edge ideals, their initial ideals, and Rees-algebra syzygies.

Abstract

We provide a closed formula for the graded Betti numbers in the linear strands of all powers of binomial edge ideals arising from closed graphs that do not have the complete graph as an induced subgraph. We show that these agree with the corresponding Betti numbers for the powers of the lexicographic initial ideal of , thereby confirming a conjecture of Ene--Rinaldo--Terai in a special case.
Paper Structure (6 sections, 7 theorems, 32 equations)

This paper contains 6 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Syzygies via relations of the Rees algebra
  5. The linear strand of $J_G^m$
  6. Acknowledgements

Key Result

Theorem 1

Let $G$ be closed and $K_4$-free graph with $e$ edges and $t$ triangles. Then for each $m\geq 1$ and $i\geq 0$ the following identities hold

Theorems & Definitions (14)

  • Theorem : \ref{['thm: linear strand']}
  • Theorem 2.1: ALL
  • Example 2.2
  • Example 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 4 more