Hilbert Coefficients and Regularity of Binomial Edge Ideals

TL;DR

This work studies binomial edge ideals $J_G$ of simple graphs by linking their Hilbert coefficients to the Castelnuovo–Mumford regularity. It proves a vertex-deletion reduction controlled by the vertex's free-clique degree: if $\mathrm{fcd}_G(v)\ge i+3$, then $e_i(S/J_G)=e_i(S/J_{G-v})$, enabling inductive computation of higher Hilbert coefficients. Using this, the paper demonstrates that for any $i\ge0$ and any pair $(r,s)$ with $r\ge2$, there exists a graph $G$ with $\mathrm{reg}(S/J_G)=r$ and $e_i(S/J_G)=s$, showing no inherent relation between regularity and Hilbert coefficients in binomial edge ideals. The authors develop explicit constructions (e.g., joins, biclaw graphs) and leverage decompositions to realize all possible pairs, highlighting rich invariant behavior beyond linear-resolution cases.

Abstract

Let $G$ be a simple graph on $n$ vertices, and let $J_G$ denotes the corresponding binomial edge ideal in $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, where $\mathbb{K}$ is a field. We show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal, thereby providing a reduction technique for computing Hilbert coefficients. As an application, for any $i\geq 0$ and a pair $(r,s)$ with $r\geq 2, s\in \mathbb{Z}$, we show that there always exists a graph $G$ such that $\mathrm{reg}(S/J_G)=r$ and $e_i(S/J_G)=s$, where $\mathrm{reg}(S/J_G)$ and $e_i(R/J_G)$ denote the Castelnuovo-Mumford regularity and the $i$-th Hilbert coefficient of $S/J_G$, respectively. In particular, this demonstrates that there is no inherent relationship between the regularity and the Hilbert coefficients for the class of binomial edge ideals.

Figures (4)

• Figure 2: The graph $B_{3,2}$
• Figure : $G$
• Figure : $G$
• Figure : $G\setminus w$

Theorems & Definitions (22)

• Theorem 2.1
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Definition 3.2
• Theorem 3.3
• proof
• Example 3.4