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Open AccessArticle

# The Regularity of Edge Rings and Matching Numbers

by Jürgen Herzog 1 and Takayuki Hibi 2,*
1
Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
2
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 39; https://doi.org/10.3390/math8010039
Received: 30 November 2019 / Revised: 18 December 2019 / Accepted: 18 December 2019 / Published: 1 January 2020
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

## Abstract

Let $K [ G ]$ denote the edge ring of a finite connected simple graph G on $[ d ]$ and $mat ( G )$ the matching number of G. It is shown that $reg ( K [ G ] ) ≤ mat ( G )$ if G is non-bipartite and $K [ G ]$ is normal, and that $reg ( K [ G ] ) ≤ mat ( G ) − 1$ if G is bipartite.
Keywords:
Let G be a finite connected simple graph on the vertex set $[ d ] = { 1 , … , d }$ and let $E ( G )$ be its edge set. Let $S = K [ x 1 , … , x d ]$ denote the polynomial ring in d variables over a field K. The edge ring of G is the toric ring $K [ G ] ⊂ S$ which is generated by those monomials $x i x j$ with ${ i , j } ∈ E ( G )$. The systematic study of edge rings originated in . It has been shown that $K [ G ]$ is normal if and only if G satisfies the odd cycle condition (, p. 131). Thus, particularly if G is bipartite, $K [ G ]$ is normal.
Let $e 1 , … , e d$ denote the canonical unit coordinate vectors of $R d$. The edge polytope is the lattice polytope $P G ⊂ R d$ which is the convex hull of the finite set . One has $dim P G = d − 1$ if G is non-bipartite and $dim P G = d − 2$ if G is bipartite. We refer the reader to (, Chapter 5) for the fundamental materials on edge rings and edge polytopes.
A matching of G is a subset $M ⊂ E ( G )$ for which $e ∩ e ′ = ∅$ for $e ≠ e ′$ belonging to M. The matching number is the maximal cardinality of matchings of G. Let $mat ( G )$ 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 $mat ( G )$. 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$reg K [ G ] ≤ mat ( G )$;
(b)
If G is bipartite, then$reg K [ G ] ≤ mat ( G ) − 1$.
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$( a 1 , … , a d ) ∈ Z d$belongs to the interior$q ( P G \ ∂ P G )$of the dilation$q P G = { q α : α ∈ P G }$, where$q ≥ 1$, of$P G$. Then$a i ≥ 1$for each$1 ≤ i ≤ d$.
Proof.
The facets of $P G$ are described in (, Theorem 1.7). When $W ⊂ [ d ]$, we write $G W$ for the induced subgraph of G on W. Since $K [ G ]$ is normal, it follows that $P G$ possesses the integer decomposition property (, p. 91). In other words, each $a ∈ q P G ∩ Z d$ is of the form
$a = ( e i 1 + e j 1 ) + ⋯ + ( e i q + e j q ) ,$
where ${ i 1 , j 1 } , … , { i q , j q }$ are edges of G.
(First Step) Let G be non-bipartite. Let $i ∈ [ d ]$. Let $H 1 , … , H s$ and $H 1 ′ , … , H s ′ ′$ denote the connected components of $G [ d ] \ { i }$, where each $H j$ is bipartite and where each $H j ′ ′$ is non-bipartite. If $s = 0$, then $i ∈ [ d ]$ is regular (, p. 414) and the hyperplane of $R d$ defined by the equation $x i = 0$ is a facet of $q P G$. Hence $a i > 0$.
Let $s ≥ 1$ and $s ′ ≥ 0$. For each $1 ≤ j ≤ s$, we write $W j ∪ U j$ for the vertex set of the bipartite graph $H j$ for which there is $a ∈ W j$ with ${ a , i } ∈ E ( G )$, where $U j = ∅$ if $H j$ is a graph consisting of a single vertex. Then $T = W 1 ∪ ⋯ ∪ W s$ is independent (, p. 414). In other words, no edge $e ∈ E ( G )$ satisfies $e ⊂ T$. Let $G ′$ denote the bipartite graph induced by T. Thus the edges of $G ′$ are ${ b , c } ∈ E ( G )$ with $b ∈ T$ and $c ∈ T ′ = U 1 ∪ ⋯ ∪ U s ∪ { i }$. Since each induced subgraph $G W j ∪ U j ∪ { i }$ is connected, it follows that $G ′$ is connected with $V ( G ′ ) = T ∪ T ′$ as its vertex set. Since the connected components of $G [ d ] \ V ( G ′ )$ are $H 1 ′ , … , H s ′ ′$, it follows that T is fundamental (, p. 415) and the hyperplane of $R d$ defined by $∑ ξ ∈ T x ξ = ∑ ξ ′ ∈ T ′ x ξ ′$ is a facet of $q P G$. Now, suppose that $a i = 0$. Since $P G$ possesses the integer decomposition property, one has $∑ ξ ∈ T a ξ = ∑ ξ ′ ∈ T ′ a ξ ′$. Hence $( a 1 , … , a d ) ∈ Z d$ cannot belong to $q ( P G \ ∂ P G )$. Thus $a i > 0$, as desired.
(Second Step) Let G be bipartite. If G is a star graph with, say, $E ( G ) = { { 1 , 2 } , { 1 , 3 } , … , { 1 , d } }$, then $P G$ can be regarded to be the $( d − 2 )$ simplex of $R d − 1$ with the vertices $( 1 , 0 , … , 0 ) , ( 0 , 1 , 0 , … , 0 ) , … , ( 0 , … , 0 , 1 )$. Thus, since each $( a 1 , … , a d ) ∈ q P G ∩ Z d$ satisfies $a 1 = q$, the assertion follows immediately. In the argument below, one will assume that G is not a star graph.
Let $i ∈ [ d ]$ and $H 1 , … , H s$ be the connected components of $G [ d ] \ { i }$. If $s = 1$, then $i ∈ [ d ]$ is ordinary (, p. 414) and the hyperplane of $R d$ defined by the equation $x i = 0$ is a facet of $q P G$. Hence $a i > 0$.
Let $s ≥ 2$. Let $W j ∪ U j$ denote the vertex set of $H j$ for which there is $a ∈ W j$ with ${ a , i } ∈ E ( G )$. Since G is not a star graph, one can assume that $U 1 ≠ ∅$. Then $T = W 2 ∪ ⋯ ∪ W s$ is independent and the bipartite graph induced by T is $G [ d ] \ ( W 1 ∪ U 1 )$. Hence T is acceptable (, p. 415) and the hyperplane of $R d$ defined by $∑ ξ ∈ W 1 x ξ = ∑ ξ ′ ∈ U 1 x ξ ′$ is a facet of $q P G$. Now, suppose that $a i = 0$. Since $P G$ possesses the integer decomposition property, one has $∑ ξ ∈ W 1 a ξ = ∑ ξ ′ ∈ U 1 a ξ ′$. Hence $( a 1 , … , a d ) ∈ Z d$ cannot belong to $q ( P G \ ∂ P G )$. Thus $a i > 0$, as required. □
We say that a finite subset $L ⊂ E ( G )$ is an edge cover of G if $∪ e ∈ L e = [ d ]$. Let $μ ( G )$ denote the minimal cardinality of edge covers of G.
Corollary 1.
When$K [ G ]$is normal, one has$q ≥ μ ( G )$if$q ( P G \ ∂ P G ) ∩ Z d ≠ ∅$.
Proof.
Since $P G$ possesses the integer decomposition property, Lemma 1 guarantees that, if $a ∈ q ( P G \ ∂ P G ) ∩ Z d$, one has $q ≥ μ ( G )$. □
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 (, p. 100) of the edge polytope $P G$, which says that the Hilbert series of $K [ G ]$ is of the form
$( h 0 + h 1 λ + ⋯ + h s λ s ) / ( 1 − λ ) ( dim P G ) + 1$
with each $h i ∈ Z$ and $h s ≠ 0$. One has
Now, Corollary 1 guarantees that
$s ≤ ( dim P G + 1 ) − μ ( G ) .$
Finally, since $μ ( G ) = d − mat ( G )$ (, Lemma 2.1), one has
$reg K [ G ] = s ≤ dim P G − ( d − 1 ) + mat ( G ) ,$
as required. □
Rafael H. Villarreal informed us that part (b) of Theorem 1 can also be deduced from (, Theorem 14.4.19).
When $K [ G ]$ is non-normal, the behavior of regularity is curious.
Proposition 1.
For given integers$0 ≤ r ≤ m$, there exists a finite connected simple graph G such that$reg K [ G ] = r$, and
Proof.
In the non-bipartite case, let H be the complete graph with $2 r$ vertices. Its matching number is r. We know from (, Corollary 2.12) that $reg K [ H ] = r$. At one vertex of H we attach a path graph of length $2 ( m − r )$ and call this new graph G. Then $mat ( G ) = m$ and $reg K [ G ] = reg K [ H ] = r$, 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 (, 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 $mat ( G ) = m + 1$ and $reg K [ G ] = reg K [ H ] = r$, 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 . Then the defining ideal of $K [ G ]$ is generated by a binomial of degree $ℓ + 3$, and hence $reg K [ G ] = ℓ + 2$, while the matching number of G is $2 + ⌈ ℓ / 2 ⌉$.
Question 1.
Let m be a positive integer, and consider the set $S m$ of finite connected simple graphs with matching number m.
• Is there a bound for $reg K [ G ]$ with $G ∈ S m$?
• If such a bound exists, is it a linear function of m?

## Author Contributions

All authors made equal and significant contributions to writing this article, and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

## Funding

Takayuki Hibi was partially supported by JSPS KAKENHI 19H00637.

## Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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