The Regularity of Edge Rings and Matching Numbers

Jürgen Herzog; Takayuki Hibi

doi:10.3390/math8010039

and

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.

Mathematics2020, 8(1), 39;https://doi.org/10.3390/math8010039

This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals

Version Notes

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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]) \leq mat (G)

if G is non-bipartite and

K [G]

is normal, and that

reg (K [G]) \leq mat (G) - 1

if G is bipartite.

Keywords:

edge ring; edge polytope; regularity; matching number

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

e_{1}, \dots, e_{d}

denote the canonical unit coordinate vectors of

R^{d}

. The edge polytope is the lattice polytope

P_{G} \subset R^{d}

which is the convex hull of the finite set

{e_{i} + e_{j} : {i, j} \in E (G)}

. 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 ([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^{'} = \emptyset

for

e \neq 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 $reg K [G] \leq mat (G)$ ;
(b): If G is bipartite, then $reg K [G] \leq mat (G) - 1$ .

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 Z^{d}

belongs to the interior

q (P_{G} \ \partial P_{G})

of the dilation

q P_{G} = {q α : α \in P_{G}}

, where

q \geq 1

, of

P_{G}

. Then

a_{i} \geq 1

for each

1 \leq i \leq d

.

Proof.

The facets of

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

P_{G}

possesses the integer decomposition property ([2], p. 91). In other words, each

a \in q P_{G} \cap Z^{d}

is of the form

a = (e_{i_{1}} + e_{j_{1}}) + \dots + (e_{i_{q}} + e_{j_{q}}),

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}^{'}, \dots, 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 \in [d]

is regular ([1], 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 \geq 1

and

s^{'} \geq 0

. For each

1 \leq j \leq 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} = \emptyset

if

H_{j}

is a graph consisting of a single vertex. Then

T = W_{1} \cup \dots \cup W_{s}

is independent ([1], p. 414). In other words, no edge

e \in E (G)

satisfies

e \subset T

. Let

G^{'}

denote the bipartite graph induced by T. Thus the edges of

G^{'}

are

{b, c} \in E (G)

with

b \in T

and

c \in T^{'} = U_{1} \cup \dots \cup U_{s} \cup {i}

. Since each induced subgraph

G_{W_{j} \cup U_{j} \cup {i}}

is connected, it follows that

G^{'}

is connected with

V (G^{'}) = T \cup T^{'}

as its vertex set. Since the connected components of

G_{[d] \ V (G^{'})}

are

H_{1}^{'}, \dots, H_{s^{'}}^{'}

, it follows that T is fundamental ([1], p. 415) and the hyperplane of

R^{d}

defined by

\sum_{ξ \in T} x_{ξ} = \sum_{ξ^{'} \in 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

\sum_{ξ \in T} a_{ξ} = \sum_{ξ^{'} \in T^{'}} a_{ξ^{'}}

. Hence

(a_{1}, \dots, a_{d}) \in Z^{d}

cannot belong to

q (P_{G} \ \partial 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

P_{G}

can be regarded to be the

(d - 2)

simplex of

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 P_{G} \cap 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] \ {i}}

. If

s = 1

, then

i \in [d]

is ordinary ([1], 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 \geq 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} \neq \emptyset

. Then

T = W_{2} \cup \dots \cup W_{s}

is independent and the bipartite graph induced by T is

G_{[d] \ (W_{1} \cup U_{1})}

. Hence T is acceptable ([1], p. 415) and the hyperplane of

R^{d}

defined by

\sum_{ξ \in W_{1}} x_{ξ} = \sum_{ξ^{'} \in 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

\sum_{ξ \in W_{1}} a_{ξ} = \sum_{ξ^{'} \in U_{1}} a_{ξ^{'}}

. Hence

(a_{1}, \dots, a_{d}) \in Z^{d}

cannot belong to

q (P_{G} \ \partial 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

μ (G)

denote the minimal cardinality of edge covers of G.

Corollary 1.

When

K [G]

is normal, one has

q \geq μ (G)

if

q (P_{G} \ \partial P_{G}) \cap Z^{d} \neq \emptyset

.

Proof.

Since

P_{G}

possesses the integer decomposition property, Lemma 1 guarantees that, if

a \in q (P_{G} \ \partial P_{G}) \cap Z^{d}

, one has

q \geq 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 ([2], 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 + \dots + h_{s} λ^{s}) / {(1 - λ)}^{(\dim P_{G}) + 1}

with each

h_{i} \in Z

and

h_{s} \neq 0

. One has

s = (\dim P_{G} + 1) - \min {q \geq 1 : q (P_{G} \ \partial P_{G}) \cap Z^{d} \neq \emptyset} .

Now, Corollary 1 guarantees that

s \leq (\dim P_{G} + 1) - μ (G) .

Finally, since

μ (G) = d - mat (G)

([3], Lemma 2.1), one has

reg K [G] = s \leq \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 ([4], Theorem 14.4.19).

When

K [G]

is non-normal, the behavior of regularity is curious.

Proposition 1.

For given integers

0 \leq r \leq m

, there exists a finite connected simple graph G such that

reg K [G] = r

, and

mat (G) = \{\begin{cases} m, if G is non - bipartite, \\ m + 1, if G is bipartite . \end{cases}

Proof.

In the non-bipartite case, let H be the complete graph with

2 r

vertices. Its matching number is r. We know from ([5], 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 ([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

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 ℓ. 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 $G \in S_{m}$ ?
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.

References

Ohsugi, H.; Hibi, T. Normal polytopes arising from finite graphs. J. Algebra 1998, 207, 409–426. [Google Scholar] [CrossRef]
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The Regularity of Edge Rings and Matching Numbers

Abstract

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Conflicts of Interest

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