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journal of Algebra
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Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases
Takayuki Hibia, Kenta Nishiyamab, Hidefumi Ohsugic *, Akihiro Shikamaa
a Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
b School of Management and Information, University of Shizuoka, Suruga-ku, Shizuoka 422-8526, Japan c Department of Mathematics, College of Science, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan
ARTICLE INFO
Article history: Received 1 February 2013 Available online 28 October 2013 Communicated by Seth Sullivant
primary 13F20
Keywords: Toric ideal Finite graph Grobner basis
ABSTRACT
Let G be a finite connected simple graph and ¡G the toric ideal of the edge ring K [G] of G. In the present paper, we study finite graphs G with the property that ¡G is generated by quadratic binomials and ¡G possesses no quadratic Grobner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for ¡G to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs G with the above property, up to 8 vertices.
© 2013 Elsevier Inc. All rights reserved.
Introduction
Let G be a finite connected simple graph on the vertex set [n] = {1, 2,...,n} with E(G) = {e1,...,ed} its edge set. (Recall that a finite graph is simple if it possesses no loop and no multiple edge.) Let K be a field and K[t] = K[t1,...,tn] the polynomial ring in n variables over K. If e = {i, j} e E(G), then te stands for the quadratic monomial tjtj e K[t]. The edge ring (see [14]) of G is the subring K [G] = K [te1, ...,ted ] of K [t]. Let K [x] = K [x1, ...,Xd] denote the polynomial ring in d variables over K with each degxi = 1 and define the surjective ring homomorphism n : K[x] ^ K[G] by setting n(xi) = tei for each 1 < i < d. The toric ideal ¡G of G is the kernel n. It is known [17, Corollary 4.3] that ¡G is generated by those binomials u — v, where u and v are monomials of K[x] with degu = degv, such that n(u) = n(v).
* Corresponding author.
E-mail addresses: hibi@math.sci.osaka-u.ac.jp (T. Hibi), k-nishiyama@u-shizuoka-ken.ac.jp (K. Nishiyama), ohsugi@rikkyo.ac.jp (H. Ohsugi), a-shikama@cr.math.sci.osaka-u.ac.jp (A. Shikama).
0021-8693/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016lj.jalgebra.2013.09.039
The distinguished properties on K[G] and ¡G in which commutative algebraists are especially interested are as follows:
(i) ¡G is generated by quadratic binomials;
(ii) K[G] is Koszul;
(iii) ¡G possesses a quadratic Grobner basis, i.e., a Grobner basis consisting of quadratic binomials.
The hierarchy (iii) ^ (ii) ^ (i) is true. However, (i) ^ (ii) is false. We refer the reader to [14] for the quick information together with basic literature on these properties. A Koszul toric ring whose toric ideal possesses no quadratic Grobner basis is given in [14, Example 2.2]. Moreover, consult, e.g., to [5, Chapter 2] for fundamental materials on Grobner bases.
We study finite connected simple graphs G satisfying the following condition:
(*) ¡G is generated by quadratic binomials and ¡G possesses no quadratic Grobner basis.
We say that a finite connected simple graphs G is (*)-minimal if G satisfies the condition (*) and if no induced subgraph H (= G) satisfies the condition (*). A (*)-minimal graph is given in [14, Example 2.1].
In the present paper, after summarizing known results on ¡G in Section 1, a nontrivial infinite series of (*)-minimal finite graphs is given in Section 2. In Section 3, we implement a combinatorial characterization for ¡G to be generated by quadratic binomials [14, Theorem 1.2] and, by means of the computer search, we classify the finite graphs G satisfying the condition (*), up to 8 vertices.
Finally, an outstanding problem is to find a finite graph G for which K [G] is Koszul, but ¡G possesses no quadratic Grobner basis. We do not know that there exists such an example in our infinite series of (*)-minimal finite graphs.
1. Known results on toric ideals of graphs
In this section, we introduce graph theoretical terminology and known results. Let G be a connected graph with the vertex set V(G) = [n] = {1, 2,..., n} and the edge set E(G). We assume that G has no loops and no multiple edges. A walk of length q of G connecting v1 e V(G) and vq+1 e V(G) is a finite sequence of the form
with each {vk, vk+1} e E(G). An even (resp. odd) walk is a walk of even (resp. odd) length. A walk r of the form (1) is called closed if vq+i = vi. A cycle is a closed walk
with q > 3 and vi = Vj for all 1 < i < j < q. A chord of a cycle (2) is an edge e e E(G) of the form e = {vi, v j} for some 1 < i < j < q with e £ E(C). If a cycle (2) is even, an even-chord (resp. odd-chord) of (2) is a chord e = {vi, vj} with 1 < i < j < q such that j — i is odd (resp. even). If e = {vi, vj} and e' = {vii, vj/} are chords of a cycle (2) with 1 < i < j < q and 1 < i' < j' < q, then we say that e and e' cross in C if the following conditions are satisfied:
(i) Either i < i' < j < j' or i' < i < j' < j.
(ii) Either {{vi, vv}, {vj, vy}} c E(C) or {{vi, vr}, {vj, vf}} c E(C).
r = (|v 1, V2}, {v2, V3},..., {Vq, Vq+i})
C = (|v 1, V2}, |V2, V3},..., |Vq, V1})
A minimal cycle of G is a cycle having no chords. If C1 and C2 are cycles of G having no common vertices, then a bridge between C1 and C2 is an edge |i, j} of G with i e V(C1) and j e V(C2).
The toric ideal ¡G is generated by the binomials associated with even closed walks. Given an even
closed walk r = (ei1, ei2,..., ei2q) of G, we write f r for the binomial
f r = f[ xi2k—1 —Yl xi2k e ¡g . k=1 k=1
It is known (see [19, Proposition 3.1], [17, Chapter 9] and [14, Lemma 1.1]) that
Proposition 1.1. Let G be a connected graph. Then, ¡G is generated by all the binomials fr, where r is an even closed walk of G. ¡n particular, ¡G = (0) if and only if G has at most one cycle and the cycle is odd.
Note that, for a binomial f e ¡G, deg( f) = 2 if and only if there exists an even cycle C of G of length 4 such that f = fC. On the other hand, a criterion for the existence of a quadratic binomial generators of ¡G is given in [14, Theorem 1.2].
Proposition 1.2. Let G be a finite connected graph. Then, ¡G is generated by quadratic binomials1 if and only if the following conditions are satisfied:
(i) ¡f C is an even cycle of G of length > 6, then either C has an even-chord or C has three odd-chords e, e' and e" such that e and e' cross in C.
(ii) ¡f C1 and C2 are minimal odd cycles having exactly one common vertex, then there exists an edge {i, j} e E(C1) U E(C2) with i e V(C1) and j e V(C2).
(iii) ¡f C1 and C2 are minimal odd cycles having no common vertex, then there exist at least two bridges between C1 and C2.
If G is bipartite, then the following is shown in [13]:
Proposition 1.3. Let G be a bipartite graph. Then the following conditions are equivalent:
(i) Every cycle of G of length > 6 has a chord.
(ii) ¡G possesses a quadratic Grobner basis.
(iii) K[G] is Koszul.
(iv) ¡G is generated by quadratic binomials.
If G is not bipartite, then the conditions (iii) and (iv) are not equivalent.
Example 1.4. (See [14, Example 2.1].) Let G be the graph in Fig. 1. Then, ¡G is generated by quadratic binomials. On the other hand, K[G] is not Koszul and hence ¡G has no quadratic Grobner bases.
1 Even if ¡G = (0), we say that "¡G is generated by quadratic binomials" and "¡G possesses a quadratic Grobner basis."
If a graph G' on the vertex set V(G') c V(G) satisfies E(G') = {{i, j} e E(G) | i, j e V(G')}, then G' is called an induced subgraph of G. The following proposition is a fundamental and important fact on the toric ideals of graphs.
Proposition 1.5. (See [12].) Let G' be an induced subgraph of a graph G. Then, K [G'] is a combinatorial pure subring ofK[G]. ¡n particular:
(i) f7c possesses a quadratic Grobner basis, then so does ¡Ci.
(ii) ¡fK[G] is Koszul, then so is K[G'].
(iii) f7G is generated by quadratic binomials, then so is ¡G>.
2. Toric ideals of the suspension of graphs
In this section, we study the existence of quadratic Grobner bases of toric ideals of the suspension of graphs.
Let G be a graph with the vertex set V(G) = [n] = {1, 2,..., n} and the edge set E(G). The suspension of the graph G is the new graph G whose vertex set is [n + 1] = V(G) U{n + 1} and whose edge set is E(G) U{{i, n + 1}| i e V(G)}. Note that, any graph G is an induced subgraph of its suspension G. The edge ideal of G is the monomial ideal ¡(G) of K[t] which is generated by {titj | {i, j} e E(G)}. See, e.g., [5, Chapter 9]. It is easy to see that the edge ring K[G] ~K[x]/^ of the suspension G of G is isomorphic to the Rees algebra
R1 (G)) = 0 I (G ) jsj = K[t1 ,...,tn, {titjs}{i, _/}eE(G)] j=0
of the edge ideal ¡(G) of G.
We now characterize graphs G such that ¡G is generated by quadratic binomials. The complementary graph G of G is the graph whose vertex set is [n] and whose edges are the non-edges of G. A graph G is said to be chordal if any cycle of length > 3 has a chord. Moreover, a graph G is said to be co-chordal if G is chordal. A graph G is called a 2K2-free graph if it is connected and does not contain two independent edges as an induced subgraph. For a connected graph G,
• G is 2K2-free — any cycle of G of length 4 has a chord in G.
• G is co-chordal ^ G is 2K2-free,
hold in general. Moreover, it is known (e.g., [1]) that Lemma 2.1. Let G be a connected graph. Then:
(i) ¡f G is co-chordal, then any cycle of G of length > 5 has a chord.
(ii) ¡f G is 2K2 -free, then any cycle of G of length > 6 has a chord.
The toric ideals ¡G of 2K2-free graphs G are studied in [15]. (In [15], 2K2-free graphs are called in a different name.) On the other hand, the edge ideals ¡(G) of 2K2-free graphs G are studied by many researchers. See, e.g., [9] and [10] together with their references and comments. (In these papers, 2K2-free graphs are called "C4-free graphs.") One can characterize the toric ideals ¡g of G that are generated by quadratic binomials in terms of 2K2-free graphs.
Theorem 2.2. Let G be a finite connected graph. Then the following conditions are equivalent:
(i) ¡g is generated by quadratic binomials;
(ii) G is 2K2-free and ¡G is generated by quadratic binomials;
(iii) G is 2K2-free and satisfies the condition (i) in Proposition 1.2.
Fig. 2. Two triangles having one common vertex.
Fig. 3. An even cycle with three odd-chords.
Proof. ((i) ^ (ii)) Suppose that ¡g is generated by quadratic binomials. Then G satisfies the conditions in Proposition 1.2. Since G is an induced subgraph of G, ¡G is generated by quadratic binomials by Proposition 1.5. Assume that G is not 2K2-free. Then, the graph in Fig. 2 is an induced subgraph of G. This contradicts that G satisfies the condition (ii) in Proposition 1.2.
((ii) ^ (i)) Suppose that G satisfies condition (ii) and that ¡g is not generated by quadratic binomials. Then the graph G does not satisfy one of the conditions in Proposition 1.2. Note that, since G satisfies the conditions in Proposition 1.2, if an even cycle or two odd cycles do not satisfy the conditions, then they have the vertex n + 1.
If an even cycle C of length > 6 has the vertex n + 1, then any other vertices of C are incident with n + 1. Thus C has an even-chord. If minimal odd cycles C1 and C2 have no common vertex and Ci contains n + 1, then n + 1 is incident with all vertices of C2. Thus, Ci and C2 has at least three bridges. Finally, suppose that minimal odd cycles C1 and C2 have exactly one common vertex v and that C1 contains n + 1. If v = n + 1, then let s (= v) be a vertex of C2. Then, since we have {n + 1, s}e E(G) \ (E(C1) U E(C2)), C1 and C2 satisfy the condition (ii) in Proposition 1.2. Let v = n + 1. Since C1 and C2 are minimal and have the vertex n + 1, the length of C1 and C2 is 3 and hence, C1 U C2 is the graph in Fig. 2. If C1 and C2 do not satisfy the condition (ii) in Proposition 1.2, then C1 U C2 is an induced subgraph of G. Thus, 2K2 is an induced subgraph of G. This is a contradiction. ((ii) ^ (iii)) It follows from Proposition 1.2.
((iii) ^ (ii)) Suppose that G satisfies the condition (i) in Proposition 1.2 and G is 2K2-free. It is enough to show that G satisfies the conditions (ii) and (iii) in Proposition 1.2. Let C1 and C2 be minimal odd cycles having exactly one common vertex v. Then there exist edges {i, j} e E(C1) and {M} e E(C2) such that v / {, j, k,£j. Since G is 2K2-free, one of {¡', k}, {i,£j, {j, k}, {j,£} belongs to E(G). Thus C1 and C2 satisfy the condition (ii) in Proposition 1.2. Let C1 and C2 be minimal odd cycles having no common vertex. Since G is 2K2-free, for each edges {i, j} e E(C1) and {k,£} e E(C2), one of {i, k}, {i,£}, {j, k}, {j,£} belongs to E(G). It then follows that there exist at least two bridges between C1 and C2. □
Example 2.3. In general, there is no implication between the two conditions: (1) ¡G is generated by quadratic binomials and (2) G is 2K2-free. In fact:
(a) Let G be the graph in Fig. 3. Then, ¡G is not generated by quadratic binomials. On the other hand, G is co-chordal (and hence 2K2-free).
(b) If G is a bipartite graph consisting of a cycle C of length 6 and a chord of C, then ¡G is generated by two quadratic binomials. On the other hand, G is not 2K2-free.
Thus, both (1) ^ (2) and (2) ^ (1) are false.
By using the theory of the Rees ring of edge ideals, we have a necessary condition for ¡g to possess a quadratic Grobner basis.
Proposition 2.4. Let G be a connected graph. ¡flG possesses a quadratic Grobner basis, then G is co-chordal.
Proof. Suppose that ¡g possesses a quadratic Grobner basis. Then, by [5, Corollary 10.1.8], each power of the edge ideal ¡(G) of G has a linear resolution. Hence, in particular, I(G) itself has a linear resolution. By Froberg's theorem [5, Theorem 9.2.3], G is co-chordal as desired. □
The converse of Proposition 2.4 is false in general. See, e.g., Example 2.9. However, if G is bipartite, then these conditions are equivalent:
Theorem 2.5. Let G be a bipartite graph. Then the following conditions are equivalent:
(i) ¡g is generated by quadratic binomials;
(ii) K[G] is Koszul;
(iii) ¡g possesses a quadratic Grobner basis;
(iv) G is 2K2-free;
(v) G is co-chordal.
Proof. First, (v) ^ (iv) is trivial. By Proposition 2.4, we have (iii) ^ (v).
((iv) ^ (i)) Suppose that G is 2K2-free. By Lemma 2.1 and Proposition 1.3, ¡G is generated by quadratic binomials. Hence (i) follows from Theorem 2.2.
((i) ^ (ii) ^ (iii)) Since G is bipartite, any odd cycle of G has the vertex n + 1. Then by [16, Proposition 5.5], there exists a bipartite graph G' such that ¡g = I g . By Proposition 1.3, I g is generated by quadratic binomials if and only if ¡g possesses a quadratic Grobner basis. Thus, three conditions (i), (ii) and (iii) are equivalent as desired.
Remark 2.6. Bipartite graphs satisfying one of the conditions in Theorem 2.5 are called Ferrers graphs (by relabeling the vertices). The edge ideal I(G) of a Ferrers graph G is well studied. See, e.g., [2] and [3].
If G is not bipartite, then the conditions (i) and (ii) in Theorem 2.5 are not equivalent. In fact,
Example 2.7. Let G be a cycle of length 5. Then G is also a cycle of length 5. Hence G is not co-chordal but 2K2-free. By Theorem 2.2 and Proposition 2.4, ¡g is generated by quadratic binomials and has no quadratic Grobner bases. Note that G is the graph in Example 1.4 and that K[G] is not Koszul.
Recall that a finite connected simple graph G is called (*)-minimal if G satisfies the condition
(*) ¡G is generated by quadratic binomials and ¡G possesses no quadratic Grobner basis
and if no induced subgraph H (= G) satisfies the condition (*). The suspension graph G given in Example 2.7 is a (*)-minimal graph. We generalize this example and give a nontrivial infinite series of (*)-minimal graphs:
Theorem 2.8. Let G be the graph on the vertex set [n] whose complement is a cycle of length n. ¡fn > 5, then G is (*)-minimal, i.e., G satisfies the following:
(i) ¡g is generated by quadratic binomials.
(ii) ¡g has no quadratic Grobner basis.
(iii) For any induced subgraph H (= G) of G, the toric ideal ¡H of H possesses a quadratic Grobner basis.
Proof. Since a cycle of length n > 5 is not chordal, (ii) follows from Proposition 2.4.
Next, we will show (iii). In [17, Theorem 9.1], a quadratic Grobner basis Gn of the toric ideal of the complete graph Kn of n vertices is constructed. In the proof, the vertices of Kn are identified with the vertices of a regular n-gon in the plane labeled clockwise from 1 to n. The Grobner basis Gn consists of quadratic binomials f such that the initial monomial of f corresponds to a pair of non-intersecting edges of Kn and the non-initial monomial of f corresponds to a pair of intersecting edges of Kn. Note that the edges {1, 2}, {2, 3},...,{n — 1, n}, {1, n} do not appear in the non-initial monomial in each binomial of Gn.
Let H be an induced subgraph of G. If H = G, then G is the cycle C = ({1, 2}, {2, 3},..., {n — 1, n}, {1, n}). By the above observation on Gn, we have a quadratic Grobner basis of ¡G by the elimination of Gn .If H = G, then H is a graph all of whose connected components are paths. Since H is a subgraph of the cycle C = ({1, 2}, {2, 3},..., {n — 1, n}, {1, n}), we have a quadratic Grobner basis of ¡H by the elimination of Gn.
Finally, we will prove the condition (ii). By the condition (iii), ¡G is generated by quadratic binomials. Moreover, since G is the cycle of length n > 5, G is 2K2-free. Thus, we have (i) by Theorem 2.2 as desired.
Even if G is co-chordal, G may be (*)-minimal:
Example 2.9. Let G be the graph whose complement is the chordal graph in Fig. 3. Then, ¡g is generated by quadratic binomials since G is co-chordal (and hence 2K2-free) and ¡G = (0). On the other hand, computational experiments in Section 3 show that G is (*)-minimal.
3. Computational experiments
In this section, we enumerate all finite connected simple graphs G satisfying the condition (*) up to 8 vertices by utilizing various software. Proposition 1.2 is a key of our enumeration method.
Proposition 1.2 gives an algorithm to determine if a toric ideal ¡G is generated by quadratic binomials. Since the criteria in Proposition 1.2 are characterized by cycles of G, we need to enumerate all even cycles and minimal odd cycles of G in order to implement the algorithm. We implement the algorithm by utilizing CyPath [18] which is a cycles and paths enumeration program implemented by T. Uno. The algorithm is used at step (2) of the following procedure to search for the graphs satisfying (*).
(1) (generating step) We use nauty [8] as a generator of all connected simple graphs with n vertices up to isomorphism.
(2) (criterion step) The criteria in Proposition 1.2 detect graphs G whose toric ideals IG are generated by quadratic binomials. These are candidates for satisfying the condition (*).
(3) (exclusion step) For each candidate G, we iterate the following computation:
(a) A new weight vector w is chosen randomly on each iteration.
(b) We compute a Grobner basis of the toric ideal IG with respect to the chosen weight vector w with Risa/Asir [11].
(c) If the Grobner basis is quadratic then the graph G is excluded from candidates.
(4) (final check step) We check the Koszul property of K[G] with Macaulay2 [4]. If it is not Koszul then IG possesses no quadratic Grobner basis. If it is indeterminable then we compute all Grobner bases by using TiGERS [6] or CaTS [7].
In our experimentation, we take 10 000 to be the number of iterations at step (3) in the case of 8 vertices. Then, there are 214 graphs as remaining candidates and we can check that 213 graphs of these are not Koszul with Macaulay2. The last one is indeterminable by computational methods in our environment. However, Theorem 2.8 tells us that it has no quadratic Grobner basis, because it is the suspension of the complement graph of a cycle whose length is 7. Therefore, we complete classification of the finite graphs with 8 vertices. Table 1 shows numbers of (1) the connected simple
Table 1
Numbers of graphs.
vertices (1) (2) (4)
3 2 2 (2) 0
4 6 6 (3) 0
5 21 20 (7) 0
6 112 95 (14) 1 (0)
7 853 568 (34) 14 (2)
8 11 117 4578 (78) 214 (51)
graphs, (2) the graphs whose toric ideals ¡G are generated by quadratic binomials (include number of zero ideals), (4) the graphs satisfying (*) (include number of the graphs which have degree 1 vertices) respectively.
We list the 14 graphs (Figs. 4-17) satisfying (*) with 7 vertices. Fig. 16 belongs to the infinite series in Theorem 2.8 and Fig. 6 is the (*)-minimal graph in Example 2.9. The list for the graphs with 8 vertices is available at
http://www2.rikkyo.ac.jp/~ohsugi/minimalexamples/
Fig. 13. Graph. Fig. 14. Graph. Fig. 15. Graph.
Fig. 16. Graph.
Fig. 17. Graph.
Acknowledgments
This research was supported by the JST (Japan Science and Technology Agency) CREST (Core Research for Evolutional Science and Technology) research project Harmony of Grobner Bases and the Modern ¡ndustrial Society in the frame of the JST Mathematics Program "Alliance for Breakthrough between Mathematics and Sciences."
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