Hilbert Coefficients and Regularity of Binomial Edge Ideals

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.