Let

G be a finite connected simple graph on the vertex set

$[d]=\{1,\dots ,d\}$ and let

$E(G)$ be its edge set. Let

$S=K[{x}_{1},\dots ,{x}_{d}]$ denote the polynomial ring in

d variables over a field

K. The

edge ring of

G is the toric ring

$K[G]\subset S$ which is generated by those monomials

${x}_{i}{x}_{j}$ with

$\{i,j\}\in E(G)$. The systematic study of edge rings originated in [

1]. It has been shown that

$K[G]$ is normal if and only if

G satisfies the odd cycle condition ([

2], p. 131). Thus, particularly if

G is bipartite,

$K[G]$ is normal.

Let

${\mathbf{e}}_{1},\dots ,{\mathbf{e}}_{d}$ denote the canonical unit coordinate vectors of

${\mathbb{R}}^{d}$. The

edge polytope is the lattice polytope

${\mathcal{P}}_{G}\subset {\mathbb{R}}^{d}$ which is the convex hull of the finite set

$\{{\mathbf{e}}_{i}+{\mathbf{e}}_{j}:\{i,j\}\in E(G)\}$. One has

$\mathrm{dim}{\mathcal{P}}_{G}=d-1$ if

G is non-bipartite and

$\mathrm{dim}{\mathcal{P}}_{G}=d-2$ if

G is bipartite. We refer the reader to ([

2], Chapter 5) for the fundamental materials on edge rings and edge polytopes.

A matching of G is a subset $M\subset E(G)$ for which $e\cap {e}^{\prime}=\varnothing $ for $e\ne {e}^{\prime}$ belonging to M. The matching number is the maximal cardinality of matchings of G. Let $\mathrm{mat}(G)$ denote the matching number of G.

When $K[G]$ is normal, the upper bound of regularity of $K[G]$ can be explicitly described in terms of $\mathrm{mat}(G)$. Our main result in the present paper is as follows:

**Theorem** **1.** Let G be a finite connected simple graph. Then

- (a)
If G is non-bipartite and$K[G]$is normal, then$\mathrm{reg}K[G]\le \mathrm{mat}(G)$;

- (b)
If G is bipartite, then$\mathrm{reg}K[G]\le \mathrm{mat}(G)-1$.

Lemma 1 stated below, which provides information on lattice points belonging to the interiors of dilations of edge polytopes, is indispensable for the proof of Theorem 1.

**Lemma** **1.** Suppose that$({a}_{1},\dots ,{a}_{d})\in {\mathbb{Z}}^{d}$belongs to the interior$q({\mathcal{P}}_{G}\backslash \partial {\mathcal{P}}_{G})$of the dilation$q{\mathcal{P}}_{G}=\{q\alpha :\alpha \in {\mathcal{P}}_{G}\}$, where$q\ge 1$, of${\mathcal{P}}_{G}$. Then${a}_{i}\ge 1$for each$1\le i\le d$.

**Proof.** The facets of

${\mathcal{P}}_{G}$ are described in ([

1], Theorem 1.7). When

$W\subset [d]$, we write

${G}_{W}$ for the induced subgraph of

G on

W. Since

$K[G]$ is normal, it follows that

${\mathcal{P}}_{G}$ possesses the integer decomposition property ([

2], p. 91). In other words, each

$\mathbf{a}\in q{\mathcal{P}}_{G}\cap {\mathbb{Z}}^{d}$ is of the form

where

$\{{i}_{1},{j}_{1}\},\dots ,\{{i}_{q},{j}_{q}\}$ are edges of

G.

**(First Step)** Let

G be non-bipartite. Let

$i\in [d]$. Let

${H}_{1},\dots ,{H}_{s}$ and

${H}_{1}^{\prime},\dots ,{H}_{{s}^{\prime}}^{\prime}$ denote the connected components of

${G}_{[d]\backslash \{i\}}$, where each

${H}_{j}$ is bipartite and where each

${H}_{{j}^{\prime}}^{\prime}$ is non-bipartite. If

$s=0$, then

$i\in [d]$ is regular ([

1], p. 414) and the hyperplane of

${\mathbb{R}}^{d}$ defined by the equation

${x}_{i}=0$ is a facet of

$q{\mathcal{P}}_{G}$. Hence

${a}_{i}>0$.

Let

$s\ge 1$ and

${s}^{\prime}\ge 0$. For each

$1\le j\le s$, we write

${W}_{j}\cup {U}_{j}$ for the vertex set of the bipartite graph

${H}_{j}$ for which there is

$a\in {W}_{j}$ with

$\{a,i\}\in E(G)$, where

${U}_{j}=\varnothing $ if

${H}_{j}$ is a graph consisting of a single vertex. Then

$T={W}_{1}\cup \cdots \cup {W}_{s}$ is independent ([

1], p. 414). In other words, no edge

$e\in E(G)$ satisfies

$e\subset T$. Let

${G}^{\prime}$ denote the bipartite graph induced by

T. Thus the edges of

${G}^{\prime}$ are

$\{b,c\}\in E(G)$ with

$b\in T$ and

$c\in {T}^{\prime}={U}_{1}\cup \cdots \cup {U}_{s}\cup \{i\}$. Since each induced subgraph

${G}_{{W}_{j}\cup {U}_{j}\cup \{i\}}$ is connected, it follows that

${G}^{\prime}$ is connected with

$V({G}^{\prime})=T\cup {T}^{\prime}$ as its vertex set. Since the connected components of

${G}_{[d]\backslash V({G}^{\prime})}$ are

${H}_{1}^{\prime},\dots ,{H}_{{s}^{\prime}}^{\prime}$, it follows that

T is fundamental ([

1], p. 415) and the hyperplane of

${\mathbb{R}}^{d}$ defined by

${\sum}_{\xi \in T}{x}_{\xi}={\sum}_{{\xi}^{\prime}\in {T}^{\prime}}{x}_{{\xi}^{\prime}}$ is a facet of

$q{\mathcal{P}}_{G}$. Now, suppose that

${a}_{i}=0$. Since

${\mathcal{P}}_{G}$ possesses the integer decomposition property, one has

${\sum}_{\xi \in T}{a}_{\xi}={\sum}_{{\xi}^{\prime}\in {T}^{\prime}}{a}_{{\xi}^{\prime}}$. Hence

$({a}_{1},\dots ,{a}_{d})\in {\mathbb{Z}}^{d}$ cannot belong to

$q({\mathcal{P}}_{G}\backslash \partial {\mathcal{P}}_{G})$. Thus

${a}_{i}>0$, as desired.

**(Second Step)** Let G be bipartite. If G is a star graph with, say, $E(G)=\{\{1,2\},\{1,3\},\dots ,\{1,d\}\}$, then ${\mathcal{P}}_{G}$ can be regarded to be the $(d-2)$ simplex of ${\mathbb{R}}^{d-1}$ with the vertices $(1,0,\dots ,0),(0,1,0,\dots ,0),\dots ,(0,\dots ,0,1)$. Thus, since each $({a}_{1},\dots ,{a}_{d})\in q{\mathcal{P}}_{G}\cap {\mathbb{Z}}^{d}$ satisfies ${a}_{1}=q$, the assertion follows immediately. In the argument below, one will assume that G is not a star graph.

Let

$i\in [d]$ and

${H}_{1},\dots ,{H}_{s}$ be the connected components of

${G}_{[d]\backslash \{i\}}$. If

$s=1$, then

$i\in [d]$ is ordinary ([

1], p. 414) and the hyperplane of

${\mathbb{R}}^{d}$ defined by the equation

${x}_{i}=0$ is a facet of

$q{\mathcal{P}}_{G}$. Hence

${a}_{i}>0$.

Let

$s\ge 2$. Let

${W}_{j}\cup {U}_{j}$ denote the vertex set of

${H}_{j}$ for which there is

$a\in {W}_{j}$ with

$\{a,i\}\in E(G)$. Since

G is not a star graph, one can assume that

${U}_{1}\ne \varnothing $. Then

$T={W}_{2}\cup \cdots \cup {W}_{s}$ is independent and the bipartite graph induced by

T is

${G}_{[d]\backslash ({W}_{1}\cup {U}_{1})}$. Hence

T is acceptable ([

1], p. 415) and the hyperplane of

${\mathbb{R}}^{d}$ defined by

${\sum}_{\xi \in {W}_{1}}{x}_{\xi}={\sum}_{{\xi}^{\prime}\in {U}_{1}}{x}_{{\xi}^{\prime}}$ is a facet of

$q{\mathcal{P}}_{G}$. Now, suppose that

${a}_{i}=0$. Since

${\mathcal{P}}_{G}$ possesses the integer decomposition property, one has

${\sum}_{\xi \in {W}_{1}}{a}_{\xi}={\sum}_{{\xi}^{\prime}\in {U}_{1}}{a}_{{\xi}^{\prime}}$. Hence

$({a}_{1},\dots ,{a}_{d})\in {\mathbb{Z}}^{d}$ cannot belong to

$q({\mathcal{P}}_{G}\backslash \partial {\mathcal{P}}_{G})$. Thus

${a}_{i}>0$, as required. □

We say that a finite subset $L\subset E(G)$ is an edge cover of G if ${\cup}_{e\in L}e=[d]$. Let $\mu (G)$ denote the minimal cardinality of edge covers of G.

**Corollary** **1.** When$K[G]$is normal, one has$q\ge \mu (G)$if$q({\mathcal{P}}_{G}\backslash \partial {\mathcal{P}}_{G})\cap {\mathbb{Z}}^{d}\ne \varnothing $.

**Proof.** Since ${\mathcal{P}}_{G}$ possesses the integer decomposition property, Lemma 1 guarantees that, if $\mathbf{a}\in q({\mathcal{P}}_{G}\backslash \partial {\mathcal{P}}_{G})\cap {\mathbb{Z}}^{d}$, one has $q\ge \mu (G)$. □

Once Corollary 1 is established, to complete the proof of Theorem 1 is a routine job on computing the regularity of normal toric rings.

**Proof of Theorem** **1.** In each of the cases (a) and (b), since the edge ring

$K[G]$ is normal, it follows that the Hilbert function of

$K[G]$ coincides the Ehrhart function ([

2], p. 100) of the edge polytope

${\mathcal{P}}_{G}$, which says that the Hilbert series of

$K[G]$ is of the form

with each

${h}_{i}\in \mathbb{Z}$ and

${h}_{s}\ne 0$. One has

Now, Corollary 1 guarantees that

Finally, since

$\mu (G)=d-\mathrm{mat}(G)$ ([

3], Lemma 2.1), one has

as required. □

Rafael H. Villarreal informed us that part (b) of Theorem 1 can also be deduced from ([

4], Theorem 14.4.19).

When $K[G]$ is non-normal, the behavior of regularity is curious.

**Proposition** **1.** For given integers$0\le r\le m$, there exists a finite connected simple graph G such that$\mathrm{reg}K[G]=r$, and **Proof.** In the non-bipartite case, let

H be the complete graph with

$2r$ vertices. Its matching number is

r. We know from ([

5], Corollary 2.12) that

$\mathrm{reg}K[H]=r$. At one vertex of

H we attach a path graph of length

$2(m-r)$ and call this new graph

G. Then

$\mathrm{mat}(G)=m$ and

$\mathrm{reg}K[G]=\mathrm{reg}K[H]=r$, as

$K[G]$ is just a polynomial extension of

$K[H]$.

In the bipartite case, let

H be the bipartite graph of type

$(r+1,r+1)$. The matching number is

$r+1$. Indeed,

$K[H]$ may be viewed as a Hibi ring whose regularity is well-known, see for example ([

6], Theorem 1.1). At one vertex of

H we attach a path graph of length

$2(m-r)$ and call this new graph

G. Then

$\mathrm{mat}(G)=m+1$ and

$\mathrm{reg}K[G]=\mathrm{reg}K[H]=r$, for the same reason as before. □

These bounds for the regularity of $K[G]$ are generally only valid if $K[G]$ is normal. Consider, for example, the graph G which consists of two disjoint triangles combined as a path of length ℓ. Then the defining ideal of $K[G]$ is generated by a binomial of degree $\ell +3$, and hence $\mathrm{reg}K[G]=\ell +2$, while the matching number of G is $2+\lceil \ell /2\rceil $.

**Question** **1.** Let m be a positive integer, and consider the set ${\mathcal{S}}_{m}$ of finite connected simple graphs with matching number m.

Is there a bound for $\mathrm{reg}K[G]$ with $G\in {\mathcal{S}}_{m}$?

If such a bound exists, is it a linear function of m?