---
abstract: 'Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[\xb, \yb] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\deg x_i = \deg y_j = 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[\xb, \yb]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i,j\}$ is an edge of $G$. The Hilbert series $H_{K[\xb, \yb]/J_G}(\lambda)$ of $K[\xb, \yb]/J_G$ is of the form $H_{K[\xb, \yb]/J_G}(\lambda) = h_{K[\xb, \yb]/J_G}(\lambda)/(1 - \lambda)^d$, where $d = \dim K[\xb, \yb]/J_G$ and where $h_{K[\xb, \yb]/J_G}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$ with each $h_i \in \ZZ$ and with $h_s \neq 0$ is the $h$-polynomial of $K[\xb, \yb]/J_G$. It is known that, when $K[\xb, \yb]/J_G$ is Cohen–Macaulay, one has $\reg(K[\xb, \yb]/J_G) = \deg h_{K[\xb, \yb]/J_G}(\lambda)$, where $\reg(K[\xb, \yb]/J_G)$ is the (Castelnuovo–Mumford) regularity of $K[\xb, \yb]/J_G$. A lot of computational experience supports the conjecture that $\reg(K[\xb, \yb]/J_G) \leq \deg h_{K[\xb, \yb]/J_G}(\lambda)$. In the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ for which $\reg(K[\xb, \yb]/J_G) = r$ and $\deg h_{K[\xb, \yb]/J_G}(\lambda) = s$ will be constructed.'
address:
- 'Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan'
- 'Kazunori Matsuda, Kitami Institute of Technology, Kitami, Hokkaido 090-8507, Japan'
author:
- Takayuki Hibi and Kazunori Matsuda
title: 'Regularity and h-polynomials of binomial edge idals'
---

Introduction {#introduction .unnumbered}
============

The binomial edge ideal of a finite simple graph was introduced in [@origin] and in [@O1] independently. (Recall that a finite graph $G$ is [*simple*]{} if $G$ possesses no loop and no multiple edge.) Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, 2, \ldots, n \}$ and $K[\xb, \yb] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\deg x_i = \deg y_j = 1$. The [*binomial edge ideal*]{} $J_G$ of $G$ is the binomial ideal of $K[\xb, \yb]$ which is generated by those binomials $x_i y_j - x_j y_i$ for which $\{i, j\}$ is an edge of $G$.

Let, in general, $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series $H_{S/I}(\lambda)$ of $S/I$ is of the form $H_{S/I}(\lambda) = (h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s)/(1 - \lambda)^d$, where each $h_i \in \ZZ$ ([@BH Proposition 4.4.1]). We say that $h_{S/I}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$ with $h_s \neq 0$ is the [*$h$-polynomial*]{} of $S/I$. Let $\reg(S/I)$ denote the ([*Castelnuovo–Mumford*]{}) [*regularity*]{} [@BH p. 168] of $S/I$. It is known (e.g., [@V Corollary B.4.1]) that, when $S/I$ is Cohen–Macaulay, one has $\reg(S/I) = \deg h_{S/I}(\lambda)$. Furthermore, in [@HM1] and [@HM2], for given integers $r$ and $s$ with $r, s \geq 1$, a monomial ideal $I$ of $S = K[x_{1}, \ldots, x_{n}]$ with $n \gg 0$ for which $\reg(S/I) = r$ and $\deg h_{S/I} (\lambda) = s$ was constructed.

Let, as before, $G$ be a finite simple graph on the vertex set $[n]$ with $d = \dim K[\xb, \yb]/J_G$ and $h_{K[\xb, \yb]/J_G}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$ the $h$-polynomial of $K[\xb, \yb]/J_G$. A lot of computational experience supports the conjecture that

\[conj\] One has $\reg(K[\xb, \yb]/J_G) \leq \deg h_{K[\xb, \yb]/J_G}(\lambda)$.

Now, in the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ on $[n]$ with $n \gg 0$ for which $\reg(K[\xb, \yb]/J_G) = r$ and $\deg h_{K[\xb, \yb]/J_G}(\lambda) = s$ will be constructed.

\[main\] Given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, there exists a finite simple graph $G$ on $[n]$ with $n \gg 0$ for which $\reg(K[\xb, \yb]/J_{G}) = r$ and $\deg h_{K[\xb, \yb]/J_{G}}(\lambda) = s$.

Proof of Theorem 0.2
====================

Our discussion starts in the computation of the regularity and the h-polynomial of the binomial edge ideal of a path graph.

\[path\] [*Let $P_{n}$ be the path on the vertex set $[n]$ with $\{1, 2\}, \{2, 3\}, \ldots, \{n - 1, n\}$ its edges. Since $K[\xb, \yb]/J_{P_{n}}$ is a complete intersection, it follows that the Hilbert series of $K[\xb, \yb]/J_{P_{n}}$ is $H(K[\xb, \yb]/J_{P_{n}}, \lambda) = (1 + \lambda)^{n - 1}/(1 - \lambda)^{n + 1}$ and that $\reg(K[\xb, \yb]/J_{P_{n}}) = \deg h_{K[\xb, \yb]/J_{P_{n}}}(\lambda) = n - 1$.* ]{}

Let $G$ be a finite simple graph on the vertex set $[n]$ and $E(G)$ its edge set. The [*suspension*]{} of $G$ is the finite simple graph $\widehat{G}$ on the vertex set $[n+1]$ whose edge set is $E(\widehat{G}) = E(G) \cup \{ \{i, n + 1\} : i \in [n] \}$. Given a positive integer $m \geq 2$, the [*m-th suspension*]{} of $G$ is the finite simple graph ${\widehat{G}}^{\,m}$ on $[n + m]$ which is defined inductively by ${\widehat{G}}^{\,m} = \widehat{{\widehat{G}}^{\,m-1}}$, where ${\widehat{G}}^{\,1} = \widehat{G}$.

\[suspension\] Let $G$ be a finite connected simple graph on $[n]$ which is not complete. Suppose that $\dim K[\xb, \yb]/J_{G} = n + 1$ and $\deg h_{K[\xb, \yb]/J_{G}} (\lambda) \geq 2$. Then $$\begin{aligned}
\reg \left(K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}} \right) = \reg(K[\xb, \yb]/J_{G}), \\
\deg h_{K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}}} (\lambda) = \deg h_{K[\xb, \yb]/J_{G}} (\lambda) + 1. \end{aligned}$$ In particular, if $ \reg(K[\xb, \yb]/J_{G}) \leq \deg h_{K[\xb, \yb]/J_{G}} (\lambda)$, then $$\reg \left(K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}} \right) < \deg h_{K[\xb, \yb]/J_{\widehat{G}}} (\lambda).$$

The suspension $\widehat{G}$ is the join product ([@SMK2 p.3]) of $G$ and $\{n + 1\}$, and $\widehat{G}$ is not complete. Hence, by virtue of [@SMK1 Theorem 2.1] and [@SMK2 Theorem 2.1 (a)], one has $$\reg \left(K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}} \right) = \max\{ \reg(K[\xb, \yb]/J_{G}), 2 \} = \reg(K[\xb, \yb]/J_{G}).$$ Furthermore, [@KS Theorem 4.6] says that $$\begin{aligned}
H\left(K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}}, \lambda \right) &=& H\left(K[\xb, \yb]/J_{G}, \lambda \right) + \frac{2\lambda + (n - 1)\lambda^{2}}{(1 - \lambda)^{n + 2}} \\
&=& \frac{h_{K[\xb, \yb]/J_{G}} (\lambda)}{(1 - \lambda)^{n + 1}} + \frac{2\lambda + (n - 1)\lambda^{2}}{(1 - \lambda)^{n + 2}} \\ 
&=& \frac{h_{K[\xb, \yb]/J_{G}} (\lambda) \cdot (1 - \lambda) + 2\lambda + (n - 1)\lambda^{2}}{(1 - \lambda)^{n + 2}}. \end{aligned}$$ Thus $\deg h_{K[\xb, \yb, x_{n+1}, y_{n+1}]/J_{\widehat{G}}} (\lambda) = \deg h_{K[\xb, \yb]/J_{G}} (\lambda) + 1$, as desired.

We are now in the position to give a proof of Theorem \[main\].

(Proof of Theorem \[main\].) Each of the following three cases is discussed.

[**(Case 1)**]{} Let $2 \leq r = s$. Let $G = P_{r + 1}$. As was shown in Example \[path\], one has $$\reg(K[\xb, \yb]/J_{G}) = \deg h_{K[\xb, \yb]/J_{G}}(\lambda) = r.$$

[**(Case 2)**]{} Let $r = 2$ and $3 \leq s$. Let $G = K_{s-1, s-1}$ denote the complete bipartite graph on the vertex set $[2s-2]$. By using [@SZ Theorem 1.1 (c) together with Theorem 5.4 (a)], one has $\reg(K[\xb, \yb]/J_{G}) = 2$ and $$\begin{aligned}
F(K[\xb, \yb]/J_{G}, \lambda) &=& \frac{1 + (2s - 3)\lambda}{(1 - \lambda)^{2s - 1}} + \frac{2}{(1 - \lambda)^{2s - 2}} - \frac{2\{1 + (s - 2)\lambda \}}{(1 - \lambda)^{s}} \\
&=& \frac{1 + (2s - 3)\lambda + 2(1 - \lambda) - 2\{1 + (s - 2)\lambda \}(1 - \lambda)^{s - 1}}{(1 - \lambda)^{2s - 1}}. \end{aligned}$$ Hence $\deg h_{K[\xb, \yb]/J_{G}} (\lambda) = s$, as required.

[**(Case 3)**]{} Let $3 \leq r < s$. Let $G = \widehat{P_{r + 1}}^{s - r}$ be the $(s - r)$-th suspension of the path $P_{r + 1}$. Applying Lemma \[suspension\] repeatedly shows $\reg (S/J_{G}) = r$ and $$\begin{aligned}
& & H(K[\xb, \yb]/J_{G}, \lambda) \\ 
& & \\
&=& \frac{(1 + \lambda)^{r} (1 - \lambda)^{s - r} + 2\lambda \left\{ \sum_{i = 0}^{s - r - 1} (1 - \lambda)^{i} \right\} + \lambda^{2} \sum_{i = 0}^{s - r - 1} (s - 1 - i)(1 - \lambda)^{i}}{(1 - \lambda)^{s + 2}} \\
& & \\
&=& \frac{(1 + \lambda)^{r} (1 - \lambda)^{s - r} + 2\lambda \cdot \frac{1 - (1 - \lambda)^{s - r}}{\lambda} + \lambda^{2} \cdot \frac{-1 + (1 - \lambda)^{s - r} + \lambda \{ s - r(1 - \lambda)^{s - r} \}}{\lambda^{2}}}{(1 - \lambda)^{s + 2}} \\
& & \\
&=& \frac{(1 + \lambda)^{r} (1 - \lambda)^{s - r} + 2\left\{1 - (1 - \lambda)^{s - r} \right\} -1 + (1 - \lambda)^{s - r} + \lambda \{s - r(1 - \lambda)^{s - r}\} }{(1 - \lambda)^{s + 2}} \\
& & \\
&=& \frac{(1 + \lambda)^{r} (1 - \lambda)^{s - r} + 1 - (1 - \lambda)^{s - r} + \lambda \{s - r(1 - \lambda)^{s - r}\}}{(1 - \lambda)^{s + 2}} \\
& & \\
&=& \frac{1 + s\lambda + (1 - \lambda)^{s - r} \{(1 + \lambda)^{r} - 1 - r\lambda \}}{(1 - \lambda)^{s + 2}}. \end{aligned}$$ Hence $\deg h_{K[\xb, \yb]/J_{G}} (\lambda) = s$, as desired.

Examples
========

A few classes of finite simple graphs which support Conjecture \[conj\] will be demonstrated.

\[cycle\] The cycle $C_n$ of length $n \geq 3$ satisfies $$\reg (K[\xb, \yb]/J_{C_{n}}) \leq \deg h_{K[\xb, \yb]/J_{C_{n}}} (\lambda).$$

Since the length of the longest induced path of $C_{n}$ is $n - 2$, it follows from [@MM Theorem 1.1] and [@regularity Theorem 3.2] that $\reg (K[\xb, \yb]/J_{C_{n}}) = n - 2$. Furthermore, [@ZZ Theorem 10 (b)] says that $$\deg h_{K[\xb, \yb]/J_{C_{n}}} = \begin{cases} 1 & (n = 3), \\ 
                                n - 1 & (n > 3).
                    \end{cases}$$ Hence the desired inequality follows.

Let $k \geq 1$ be an integer and $p_{1}, p_{2}, \ldots, p_{k}$ a sequence of positive integers with $p_{1} \geq p_{2} \geq \cdots \geq p_{k} \geq 1$ and $p_{1} + p_{2} + \cdots + p_{k} = n$. Let $V_{1}, V_{2}, \ldots, V_{k}$ denote a partition of $[n]$ with each $|V_{i}| = p_{i}$. In other words, $[n] = V_{1} \sqcup V_{2} \sqcup \cdots \sqcup V_{k}$ and $V_{i} \cap V_{j} = \emptyset$ if $i \neq j$. Suppose that $$V_{i} = \left\{\sum_{j = 1}^{i - 1} p_{j} + 1, \ \sum_{j = 1}^{i - 1} p_{j} + 2, \ \ldots,  \sum_{j = 1}^{i - 1} p_{j} + p_{i} - 1,  \sum_{j = 1}^{i} p_{j} \right\}$$ for each $1 \leq i \leq k$. The [*complete multipartite graph*]{} $K_{p_{1}, \ldots, p_{k}}$ is the finite simple graph on the vertex set $[n]$ with the edge set $$E\left(K_{p_{1}, \ldots, p_{k}}\right) = \{ \, \{k, \ell\} \, : \, k \in V_{i}, \, \ell \in V_{j}, \, 1 \leq i < j \leq k \, \}.$$

\[completemultipartite\] The complete multipartite graph $G = K_{p_{1}, \ldots, p_{k}}$ satisfies $$\reg (K[\xb, \yb]/J_{G}) \leq \deg h_{K[\xb, \yb]/J_{G}} (\lambda).$$

We claim $\reg (K[\xb, \yb]/J_{G}) \leq \deg h_{K[\xb, \yb]/J_{G}} (\lambda)$ by induction on $k$. If $k = 1$, then $G = K_{p_{1}}$ is the complete graph and $\reg (K[\xb, \yb]/J_{G}) = \deg h_{K[\xb, \yb]/J_{G}} (\lambda) = 1$.

Let $k > 1$. If $p_{k} = 1$, then $G = \widehat{G'}$, where $G' = K_{p_{1}, \ldots, p_{k - 1}}$. Lemma \[suspension\] as well as the induction hypothesis then guarantees that $\reg (K[\xb, \yb]/J_{G}) \leq \deg h_{K[\xb, \yb]/J_{G}} (\lambda)$. Hence one can assume that $p_{k} > 1$. In particular $G$ is not complete. It then follows from [@SMK2 Theorem 2.1(a)] that $\reg (K[\xb, \yb]/J_{G}) = 2$. Furthermore, [@KS Corollary 4.14] says that $$\deg h_{K[\xb, \yb]/J_{G}} (\lambda) = \begin{cases} n - p_{k} + 1 & (2p_{1} < n + 1), \\ 
                                2p_{1} - p_{k} & (2p_{1} \geq n + 1).
                    \end{cases}$$ Since $k > 1$ and $p_{k} > 1$, one has $\deg h_{K[\xb, \yb]/J_{G}} (\lambda) \geq n - p_{k} + 1 \geq p_{1} + 1 \geq 3$. Thus the desired inequality follows.

Let $t \geq 3$ be an integer and $K_{1, t}$ the complete bipartite graph on $\{1, v_{1}, \ldots, v_{t}\}$ with the edge set $E(K_{1, t}) = \{ \, \{1, v_{i}\} \, : \, 1 \leq i \leq t \, \}$. Let $p_{1}, p_{2}, \ldots, p_{t}$ be a sequence of positive integers and $P^{(i)}$ the path of length $p_{i}$ on the vertex set $\{w_{i, 1}, w_{i, 2}, \ldots, w_{i, p_{i} + 1} \}$ for each $1 \leq i \leq t$. Then the [*t-starlike graph*]{} $T_{p_{1}, p_{2}, \ldots, p_{t}}$ is defined as the finite simple graph obtained by identifying $v_{i}$ with $w_{i, 1}$ for each $1 \leq i \leq t$. Thus the vertex set of $T_{p_{1}, p_{2}, \ldots, p_{t}}$ is $$\{1\} \cup \bigcup_{i = 1}^{t} \{w_{i, 1}, w_{i, 2}, \ldots, w_{i, p_{i} + 1} \}$$ and its edge set is $$E(T_{p_{1}, p_{2}, \ldots, p_{t}}) = \bigcup_{i = 1}^{t} \left\{ \{ w_{i, j}, w_{i, j + 1} \} \mid 0 \leq j \leq p_{i} \right\},$$ where $w_{i, 0} = 1$ for each $1 \leq i \leq t$.

\[starlike\] The $t$-starlike graph $G = T_{p_{1}, p_{2}, \ldots, p_{t}}$ satisfies $$\reg (K[\xb, \yb]/J_{G}) < \deg h_{K[\xb, \yb]/J_{G}} (\lambda).$$

It follows from [@JNRR Corollary 3.4 (2)] that $\reg(K[\xb, \yb]/J_{G}) = 2 + \sum_{i = 1}^{t} p_{i}$. Furthermore, [@SZ Theorem 5.4 (a)] guarantees that $$\begin{aligned}
F(K[\xb, \yb]/J_{K_{1, t}}) &=& \frac{1}{(1 - \lambda)^{2t}} - \frac{1 + (t - 1)\lambda}{(1 - \lambda)^{t + 1}} + \frac{1 + t\lambda}{(1 - \lambda)^{t + 2}} \\
& & \\
&=& \frac{1 - \{ 1 + (t - 1) \lambda \}(1 - \lambda)^{t - 1} + (1 + t\lambda)(1 - \lambda)^{t - 2}}{(1 - \lambda)^{2t}}  \\ 
& & \\
&=& \frac{1 + (1 - \lambda)^{t - 2} \left\{ 2\lambda + (t - 1)\lambda^{2} \right\}}{(1 - \lambda)^{2t}} \end{aligned}$$ Hence, by virtue of [@KS Corollary 3.3], one has $$h_{K[\xb, \yb]/J_{G}} (\lambda) = \left[1 + (1 - \lambda)^{t - 2} \left\{ 2\lambda + (t - 1)\lambda^{2} \right\} \right] \cdot (1 - \lambda)^{\sum_{i = 1}^{t} p_{i}}.$$ Thus $$\deg h_{K[\xb, \yb]/J_{G}} (\lambda) = t + \sum_{i = 1}^{t} p_{i} > 2 + \sum_{i = 1}^{t} p_{i} = \reg(K[\xb, \yb]/J_{G}),$$ as required.

*Let $m \geq 0$ be an integer and $G_{m}$ the finite simple graph on the vertex set $[m + 9]$ drawn below:*

@ (0,0);(30,0) \*; @ (0,0);(80, 10) \*++!D[8]{} \* = “A”; @[-]{} “A”;(50, 0) \*++!U[1]{} \* = “B”; @[-]{} “B”;(60, 0) \*++!U[2]{} \* = “C”; @[-]{} “C”;(70, 0) \*++!U[3]{} \* = “D”; @[-]{} “D”;(80, 0) \*++!LD[4]{} \* = “E”; @[-]{} “E”;(90, 0) \*++!U[5]{} \* = “F”; @[-]{} “F”;(100, 0) \*++!U[6]{} \* = “G”; @[-]{} “G”;(110, 0) \*++!U[7]{} \* = “H”; @[-]{} “A”;“H”; @[-]{} “A”;“E”; @[-]{} “E”;(80, -8) \*++!L[9]{} \* = “I”; @[-]{} “I”;(80, -12) = “J”; @[.]{} “J”;(80, -16) = “K”; @[-]{} “K”;(80, -20) \*++!L[m + 9]{} \* = “L”;

Then $K[\xb, \yb]/J_{G_{m}}$ is not unmixed. In fact, for each subset $S \subset [m + 9]$, we define $$P_{S} = \left( \bigcup_{i \in S} \{ x_{i}, y_{i} \}, J_{\tilde{G}_{1}}, \ldots, J_{\tilde{G}_{c(S)}} \right),$$ where $G_{1}, \ldots, G_{c(S)}$ are connected components of $G_{[m + 9] \setminus S}$ and where $\tilde{G}_{1}, \ldots, \tilde{G}_{c(S)}$ is the complete graph on the vertex set $V(G_{1}), \ldots, V\left(G_{c(S)}\right)$, respectively. It then follows from [@origin Lemma 3.1 and Corollary 3.9] that $P_{\emptyset}$ and $P_{\{ 3, 8 \}}$ are minimal primes of $J_{G_{m}}$ and that $\height P_{\emptyset} = m + 8 < \height P_{ \{3, 8\} } = m + 9$. Thus $K[\xb, \yb]/J_{G_{m}}$ is not unmixed. In particular, $K[\xb, \yb]/J_{G_{m}}$ is not Cohen-Macaulay. However, one has $\reg (K[\xb, \yb]/J_{G_{m}}) = \deg h_{K[\xb, \yb]/J_{G_{m}}} (\lambda) = m + 6$.

[**Acknowledgements**]{} The first author is partially supported by JSPS KAKENHI 26220701. The second author is partially supported by JSPS KAKENHI 17K14165.

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