Regularity and h-polynomials of binomial edge ideals

Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[X, Y]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[X, Y]/J_G}(λ)$ of $K[X, Y]/J_G$ is of the form $H_{K[X, Y]/J_G}(λ) = h_{K[X, Y]/J_G}(λ)/(1 - λ)^d$, where $d = \mathrm{dim} K[X, Y]/J_G$ and where $h_{K[X, Y]/J_G}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with each $h_i \in \mathbb{Z}$ and with $h_s \neq 0$ is the $h$-polynomial of $K[X, Y]/J_G$. It is known that, when $K[X, Y]/J_G$ is Cohen-Macaulay, one has $\mathrm{reg}(K[X, Y]/J_G) = \mathrm{deg} h_{K[X, Y]/J_G}(λ)$, where $\mathrm{reg}(K[X, Y]/J_G)$ is the (Castelnuovo-Mumford) regularity of $K[X, Y]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[X, Y]/J_G) = r$ and $\mathrm{deg} h_{K[X, Y]/J_G}(λ) = s$ will be constructed.

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