Complete intersection toric ideals of oriented graphs - Cinvestav

Morfismos, Vol. 9, No. 2, 2005, pp. 71–82

Complete intersection toric ideals of oriented graphs∗ Enrique Reyes

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Abstract Let G be a connected simple graph. We prove the existence of an orientation O of G such that the toric ideal asociated to the digraph D = (G, O) is a complete intersection.

2000 Mathematics Subject Classification: 13H10, 05C85, 05C20. Keywords and phrases: toric ideal, oriented graph, complete intersection, vector matroid.

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Introduction

Let G be a connected simple graph and let O be an orientation of G. In Section 2, we compare the circuits and the bases of the matroids associated to G and to D = (G, O). We study when the toric ideal associated to D is a complete intersection and show that this ideal is generated by the binomials corresponding to primitive cycles. In Section 3 we construct an orientation O and a spanning tree τ of G, such that the toric ideal PD of D, is a complete intersection. In particular we prove that the binomials corresponding to the principal cycles c(τ, fi ) with fi ∈ E(D) \ E(τ ) are a Gröbner basis of PD , where c(τ, fi ) is the unique cycle of the subgraph τ ∪ {fi }. As usual we are denoting the edge set of D by E(D) and the vertex set of D by V (D). We prove that if D is an acyclic oriented graph with a spanning directed path, then PD is a complete intersection minimally generated by a Gröbner ∗

This paper is part of the author’s Ph.D. thesis presented at the Department of Mathematics of CINVESTAV-IPN under the direction of Prof. Rafael H. Villarreal. 1 Partially supported by COFAA-IPN, and CONACyT grant 49251-F.

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basis. This complements a result of Ishizeki [3] showing that all acyclic complete oriented graphs have this property.

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Toric ideals of oriented graphs

Let G be a connected simple graph with n vertices and q edges, and let O be an orientation of G. Thus D = (G, O) is an oriented graph. In particular D is a digraph. To each oriented edge e = (xi , xj ) of D, we associate the vector ve defined as follows: the ith entry is −1, the jth entry is 1, and the remaining entries are zero. The incidence matrix AD of D is the n × q matrix with entries in {0, ±1} whose columns are the vectors of the form ve , with e an edge of D. For simplicity of notation we set A = AD . The set of column vectors of A will be denoted by A = {v1 , . . . , vq }. It is well known [4] that A defines a matroid M [A] on A = {v1 , . . . , vq } over the field Q of rational numbers, which is called the vector matroid of A, whose independent sets are the independent subsets of A. Definition 2.1 A minimal dependent set or circuit of M [A] is a dependent set all of whose proper subsets are independent. A subset B of A is called a basis of M [A] if B is a maximal independent set. Lemma 2.2 The circuits of M [A] are precisely the cycles of G, A is totally unimodular, and rank(A) = n − 1. Proof:

It follows from [1, pp. 343-344] and [6, p. 274].

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In contrast if we consider the {0, 1} incidence matrix of a simple graph, we have the following description of the circuits and bases. Proposition 2.3 ([8]) If AG is the incidence matrix of a connected simple graph G, then the circuits of the vector matroid M [AG ] consists of even cycles, or two edge disjoint odd cycles meeting at exactly one vertex, or two vertex disjoint odd cycles joined by an arbitrary path. Proposition 2.4 ([8]) If G is bipartite, then the bases of M [AG ] are the spanning tree. If G is not bipartite, then the basis of M [AG ] are the spaning tree subgraphs of G whose connected components are unicyclic graphs with an odd cycle.

Complete intersection toric ideals

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Definition 2.5 An elementary vector of ker(A) is a vector 0 ̸= α in ker(A) whose support is minimal with respect to inclusion, i.e., supp(α) does not properly contain the support of any other nonzero vector in ker(A). A circuit of ker(A) is an elementary vector of ker(A) with relatively prime integral entries. It is interesting to observe that there is a one to one correspondence Circuits of ker(A) −→

Circuits of M [A] = cycles of G

given by α = (α1 , . . . , αq ) −→ C(α) = {vi | i ∈ supp(α)}. Thus the set of circuits of the kernel of A is the algebraic realization of the set of circuits of the vector matroid M [A]. Consider the monomial subring ±1 k[D] := k[xv1 , . . . , xvq ] ⊂ k[x±1 1 , . . . , xn ]

associated to the digraph D = (G, O). There is an epimorphism of k-algebras φ: B = k[t1 , . . . , tq ] −→ k[D], ti 7−→ xvi , where B is a polynomial ring. The kernel of φ, denoted by PD , is called the toric ideal of k[D]. Notice that PD is no longer a graded ideal of B, see Proposition 2.12. The toric ideal PD is a prime ideal of height q − n + 1 generated by binomials (see Lemma 2.2). Thus any minimal generating set of PD must have at least q − n + 1 elements, by the principal ideal theorem. Definition 2.6 The toric ideal PD is called a complete intersection if PD can be generated by q − n + 1 polynomials. If 0 ̸= α ∈ ker(A) ∩ Zn we associate the binomial tα = tα + − tα − . Notice that tα ∈ PD .

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Given a cycle c of D, we split c in two disjoint sets of edges c+ and c− , where c+ is oriented clockwise and c− = c \ c+ . The binomial ∏ ∏ tc = ti − ti vi ∈c+

vi ∈c−

belongs to PD . If c+ = ∅ or c− = ∅ we set ∏ ∏ ti = 1. ti = 1 or vi ∈c+

vi ∈c−

Proposition 2.7 PD is generated by the set of all binomials tc such that c is a cycle of D and this set is a universal Gr¨ obner basis. Proof: Let UD be the set of all binomials of the form tα such that α is a circuit of ker(A). Since A is totally unimodular, by [7, Proposition 8.11], the set UD form a universal Gröbner basis of PD . Notice that the circuits of ker(A) are in one to one correspondence with the circuits of the vector matroid M [A]. To complete the proof it suffices to observe that the circuits of M [A] are precisely the cycles of G, see Lemma 2.2. 2 Proposition 2.8 Let c = {x1 , x2 , . . . , xr , x1 } be a circuit of D. Suppose that (xi , xj ) or (xj , xi ) is an edge of D, with i + 1 < j. Then tc is a linear combination of tc1 and tc2 , where c1 = {x1 , x2 , . . . , xi , xj , xj+1 , . . . , xr , x1 } and c2 = {xi , xi+1 , . . . , xj , xi }. Proof: Suppose without loss of generality that vk = (xi , xj ) is the edge of D. Set tc1 = tα+ − tα− and tc2 = tβ+ − tβ− . We can suppose that vk ∈ c1+ ∩ c2+ because if it is false then we multiply tc1 or tc2 by −1. Since tk |tα+ and tk |tβ+ then (β ) (α ) (β ) ( ) t + t + t + α+ − tα− ) − tα+ (tβ+ − tβ− ) t − t = (t c c 1 2 tk tk tk ( αtk ) (β ) t + t + β α − − = t − tk t = tγ1 − tγ2 tk is in k[t1 , . . . , tq ], where γ1 = (α+ − ek ) + β− and γ2 = (β+ − ek ) + α− . Then tγ1 is the product of the edges of (c1+ \ {tk }) ∪ c2− , but these are the edges of c+ . In a similar form tγ2 is the product of edges of c− . Then tc = tγ1 − tγ2 . 2 Definition 2.9 A cycle c of D is called primitive if c is an induced subgraph of D.

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As an immediate consequence of Propositions 2.7 and 2.8 we get: Corollary 2.10 PD is generated by the set of binomials corresponding to primitive cycles. We say that a cycle c of D is directed if all the arrows of D are oriented in the same direction. If D does not have directed cycles, we say that D is acyclic. The following is well known: Proposition 2.11 ([2]) D is acyclic if and only if there is a linear ordering x1 , . . . , xn of the vertex set, such that every edge of D has the form (xi , xj ) with i < j. The ordering of the last proposition is called topological ordering or topological sort. The following result is not hard to prove. Proposition 2.12 If D has a topological ordering, then PD is generated by homogeneous binomials with respect to the order induced by degree(tk ) = j − i, where tk maps to xi xj and (xi , xj ) is an edge. Corollary 2.13 If D is an arbitrary acyclic directed graph, then PD is a complete intersection if and only if PD is generated by q − n + 1 binomials corresponding to primitive cycles. Proof:

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It follows from Corollary 2.10 and Proposition 2.11.

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Complete intersection generator tree of G

The aim here is to show the existence of an orientation O of G such that the toric ideal of D = (G, O) is a complete intersection. We are going to construct a proper nested sequence A1 , . . . , Am of subtrees of G label by V (Aj ) = {y1j , . . . , yrjj } such that Am is a spanning tree of G, with certain special properties. Let A1 be a path of G maximal 1 {y1 , y21 , . . . , yr11 }. We define

with respect to |V (A1 )|. Set V (A1 ) =

i1 = max{u ∈ N|NG (y11 , . . . , yu1 ) ⊂ V (A1 )}, where NG (B) stands for the neighbor set of B. If i1 < |V (A1 )| we define a1 = yi11 +1 . If i1 = |V (A1 )|, we set m = 1. By induction we define Ai

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as follows. Suppose Aj has been defined, where V (Aj ) = {y1j , . . . , yrjj }. We define ij = max{u ∈ N|NG (y1j , . . . , yuj ) ⊂ V (Aj )}. If ij < |V (Aj )|, we define aj = yijj +1 . Let Lj be a maximal path with respect to the cardinality, with the properties V (Lj ) ∩ V (Aj ) = {aj } and V (Lj ) = {z1j , z2j , . . . , zsjj = aj }, the final vertex of Lj is aj . We define Aj+1 in the following way: V (Aj+1 ) = V (Aj ) ∪ V (Lj ) = {y1j+1 , . . . , yrj+1 } where j +sj −1

(1)

yij+1

 j  if i ≤ ij ,  yi j z if ij + 1 ≤ i ≤ ij + sj , = i−ij   yj i−sj +1 if ij + sj + 1 ≤ i ≤ rj + sj − 1

with E(Aj+1 ) = E(Aj ) ∪ E(Lj ), and rj+1 = rj + sj − 1. If ij = |V (Aj )|, we set m = j. Lemma 3.1 ik+1 > ik for 1 ≤ k ≤ m − 1. Proof: By construction yik+1 = yik for i ≤ ik (see Eq.(1)). If ik+1 < ik , then ak+1 = yik+1 = yikk+1 +1 . By the definition of ik+1 , there exists k+1 +1 x ∈ V (G) \ V (Ak+1 ) ⊂ V (G) \ V (Ak ) such that {x, yik+1 } ∈ E(G). But ik+1 +1 ≤ ik , a contradiction by the k+1 +1 definition of ik . Notice that ik+1 can not be equal to ik by construction of Lk . 2 Suppose that the process finish in the step m. Lemma 3.2 Ai is a tree, for 1 ≤ i ≤ m. Proof: By induction over i. For i = 1 is clear. Suppose Ai is a tree. Recall that Li is a tree and V (Li ) ∩ V (Ai ) = {ai }. On the other hand V (Ai+1 ) = V (Ai ) ∪ V (Li ) and E(Ai+1 ) = E(Ai ) ∪ E(Li ), then Ai+1 is connected and does not have cycles. 2 Lemma 3.3 Am is a spanning tree of G.


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Proof: By the construction im = rm . Hence if {x, yum } ∈ E(G), for some u, then x ∈ V (Am ). Suppose that a ∈ V (G) \ V (Am ). G is connected, then there exists a path L = {b1 = y1m , b2 , . . . , bl = a}. Let j = max{u ∈ N|bu ∈ Am }. We have that j < l, then {bj , bj+1 } ∈ E(G) and bj ∈ V (Am ), but bj+1 is not in V (Am ) this is a contradiction. Then V (Am ) = V (G). 2 In Lemma 3.3 we are essentially proving the following general fact. Lemma 3.4 If H is a subgraph of a connected graph G and NG (V (H)) is contained in V (H), then V (G) = V (H).

Orientation of the tree Am and the graph G Let τ = (Am , O) be the digraph obtained from Am with the orientation (yim , yjm ) ∈ E(τ ) if and only if {yim , yjm } ∈ E(Am ) and j > i. We have that V (G) = V (Am ) = {y1m , y2m , . . . , yrmm } and we will orient G to obtain the digraph D = (G, O) in the following form (yim , yjm ) ∈ E(D) if and only if {yim , yjm } ∈ E(G) and j > i. Example 3.5 The construction of Am and the orientation O of G, is illustrated below. r y11 @ @ R @ 1 y r 4 @r y21 y51 r @ r rP @ PP 6 P @r e yP 1 P PP r 3 r r @ @ @r r r @ 2 z2 rP @ PP r qP P @r 2 P z3P PP r z12 r6

r @ @ @r r

r @ rP @ r z1 PP 1 @r P ? 1 P z3P iP r z21 P r r y13 @ @ R @ 3

r r y8 @r y23 @ r y3 y63 rP @ P qP 6 3 P ? @r e P 3 PP 3 6 i y P r y43 y5 r 7 y93

r y12 @ @ R @ 2 y y72 r r 6 @r y22 @ r y2 rP @ PP 6 3 P @r e ? PP P iP r y42 y52 r r @ R @ @ @r r r @ r rP @ P qP ) 6 I @ P ? @ r P PP 6 1 i Pr r

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Lemma 3.6 Let ylk1 and ylk2 be vertices in Ak such that ylk1 = yjr1 and ylk2 = yjr2 with k ≤ r ≤ m. If l1 < l2 , then j1 < j2 . Proof: If r = k the lemma is clear. In other case, is suficient to prove , then q2 > q1 . If l2 ≤ ik , then q2 = l2 and ylk2 = yqk+1 that if ylk1 = yqk+1 2 1 and q1 = l1 . If l1 ≤ ik ≤ l2 , then q1 = l1 and q2 = l2 + sk − 1 ≥ l2 . Thus q2 > q1 . If ik < l1 then q1 = l1 + sk − 1 and q2 = l2 + sk − 1. Thus q2 > q 1 . 2 Lemma 3.7 {yikk +1 , yikk +2 , . . . , yrkk } is a path in Ak for 1 ≤ k ≤ m. Proof: By induction over k. For k = 1 the lemma is clear . Suppose the result for k. We can suppose that k < m. We have that yik+1 ∈ V (Ak+1 ) = V (Ak ) ∪ V (Lk ). k+1 If yik+1 ∈ V (Ak ), then there exists j such that yik+1 = yjk . Thus k+1 +1 k+1 +1 j ≥ ik + 1 because if j ≤ ik by Eq.(1) j = ik+1 + 1, then ik > ik+1 and this is false by Lemma 3.1. Set L = {yik+1 , yik+1 , . . . , yrk+1 } = {yikk+1 −sk , . . . , yrkk }. k+1 k+1 +1 k+1 +2 Since L is a subwalk of {yikk +1 , . . . , yrkk } and the last is a path by induction hypotesis, then L is a path. Thus yik+1 ∈ / V (Ak ), and k+1 +1 yik+1 = zjk for some j and zjk ∈ V (Lk ). Let k+1 +1 B = {zjk , . . . , zskk } = {yik+1 , yik+1 , . . . , yik+1 } k+1 +1 k+1 +2 k +sk be the path in Ak that join zj and yikk +1 = yik+1 . By the induction k +sk hypothesis , . . . , yrk+1 C = {yikk +1 , . . . , yrkk } = {yik+1 } k +sk −1 k +sk is a path. As B ⊆ V (Ak ) and C ⊆ V (Lk ), then B ∩ C = {yik+1 } and k +sk k+1 k+1 2 {yik+1 +1 , . . . , yrk+1 } is a path. Proposition 3.8 Let yik and yjk be vertices in Ak with j > i such that {yik , yjk } ∈ E(G) \ E(Ak ). If F = {ylk1 = yik , ylk2 , . . . , ylkr = yjk } is the walk that join yik and yjk in Ak , then l1 < l2 < · · · < lr .

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Proof: By induction over k. If k = 1 is clear. Suppose that is true for k = q, and we will prove the result for k = q + 1. Case (a): yiq+1 , yjq+1 ∈ V (Aq ). As V (Aq ) ⊆ V (Aq+1 ) and Aq+1 , Aq are trees, if F ′ is the path that join yiq+1 and yjq+1 in Aq , then F ′ = F . If F ′ = {ylq′ = yiq+1 , ylq′ = ylq+1 , . . . , ylq′ = yjq+1 } 2 1

l1′

2

r

l2′ .

by Lemma 3.6 < Then by the induction hypotesis l1′ < l2′ < · · · < lr′ . And by Lemma 3.6 l1 < l2 < · · · < lr . Case (b): yiq+1 , yjq+1 are not in V (Aq ). As V (Aq+1 ) = V (Aq )∪V (Lq ) q then yiq+1 , yjq+1 ∈ V (Lq ) = {z1q , z2q , ..., zsqq } so yiq+1 = zi−i and yjq+1 = q q zj−i . The path in Aq+1 that join yiq+1 and yjq+1 is q q+1 q q q {yiq+1 , yi+1 , . . . , yjq+1 } = {zi−i , zi−i , . . . , zj−i }. q q +1 q

Case (c): |V (Aq ) ∩ {yiq+1 , yjq+1 }| = 1. If yiq+1 ∈ V (Aq ) then yjq+1 ∈ q V (Aq+1 ) \ V (Aq ), and yjq+1 = zj−i ∈ V (Lq ). As j > i then i ≤ iq q and yiq+1 = yiq . But {yiq+1 , yjq+1 } ∈ E(G), then by the definition of iq , yjq+1 ∈ V (Aq ), this is a contradiction. So yjq+1 ∈ V (Aq ) and yiq+1 ∈ q q V (Aq+1 ) \ V (Aq ) thus yiq+1 = zi−i with zi−iq ∈ V (Lq ) and i < iq + sq . q q This implies yjq+1 = aq = yiqq +1 or yjq+1 = yj−s . In the first case q +1 q {yiq+1 , . . . , yjq+1 } = {zi−iq , . . . , zsqq = aq }

is the path that join yiq+1 and yjq+1 . In the second case j ≥ iq + sq + 1 then j − sq + 1 ≥ iq + 2, we have that {yiqq +1 , yiqq +2 , . . . , yrqq } = {yiq+1 , yiq+1 , . . . , yrq+1 } q +sq −1 q +sq q +sq +1 and by the Lemma 3.7 is a path. As j ≥ iq + sq + 1, then {yiq+1 ,..., q +sq yjq+1 } is a path, and q {yiq+1 , . . . , yiq+1 } = {zi−iq , . . . , zsqq } q +sq

is another path and their intersection is {yiq+1 }. Since q +sq q+1 {yiq+1 , yi+1 , . . . , yiq+1 , . . . , yjq+1 } q +sq

is the path that join to yiq+1 and yjq+1 in Aq+1 .

2

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Let τ be the tree defined after Lemma 3.4. For each fi ∈ E(D)\E(τ ) the unique cycle of the subgraph τ ∪ {fi } is denoted by c(τ, fi ). Notice that Proposition 3.8 is equivalent to the following. Proposition 3.9 For each fi ∈ E(D) \ E(τ ) all edges of c(τ, fi ) \ {fi } are oriented in the same direction and fi is oriented in opposite direction. ( ) Theorema 3.10 PD = {tc(τ,fi ) |fi ∈ E(D) \ E(τ )} . Proof: Set E(D) \ E(τ ) = {f1 , . . . , fq−n+1 }. Suppose without loss of generality that t1 , . . . , tq−n+1 are associated to f1 , . . . , fq−n+1 respectively. By the Proposition 3.8 tc(τ,fi ) = ti − tβi , where tβi is a product of variables associated to edges in τ . Let I be the ideal generated by the set {tc(τ,fi ) |fi ∈ E(D) \ E(τ )} in B = k[t1 , . . . , tq ]. Let h = tα − tβ be a binomial in PD . Thus ti = tβi in B/I for i = 1, . . . , q − n + 1. Then h = tγ − tω , where tγ and tω are product of variables asociated to τ . As I ⊆ PD then tγ − tω ∈ PD = ker(φ). But τ is a tree thus tγ = tω and h = 0, in B/I. Since PD is generated by binomials, PD = I. 2 Corollary 3.11 Let D be the directed graph constructed above. Then PD is generated by q − n + 1 primitive cycles, in particular PD is a complete intersection. Corollary 3.12 PD is homogeneous ideal with the grading induced by deg(tk ) = j − i, where tk maps to xi xj and (xi , xj ) ∈ E(D). Proof: By construction D is acyclic. Thus we may apply Propositions 2.11 and 2.12. 2

Definition 3.13 A tournament D is a complete graph Kn with an orientation. Proposition 3.14 ([2]) If D is a tournament, then D has a spanning path. Proposition 3.15 If D is an acyclic tournament, then PD is a complete intersection minimally generated by a Gr¨ obner basis.


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Proof: Let δ be a spanning path of D, i.e. δ = (x1 , x2 , . . . , xn ) and (xi , xi+1 ) ∈ E(D) for all i < n. Since D is acyclic it follows that PD = ({tc(δ,fi ) |fi ∈ E(D) \ E(δ)}), where for each fi ∈ E(D) \ E(δ), the unique cycle of the subgraph δ ∪ {fi } is denoted by c(δ, fi ). See the proof of Theorem 3.10. If < is the lexicographical order induced by any linear ordening of t1 , . . . , tq such that tj < ti if fi ∈ E(D) \ E(δ) and fj ∈ E(δ). Then it is seen that {tc(δ,fi ) |fi ∈ E(D) \ E(δ)} is a Gröbner basis. 2 Similarly we can prove the following: Proposition 3.16 If D is an acyclic oriented graph with a spanning directed path, then PD is a complete intersection minimally generated by a Gr¨ obner basis. A problem here is to characterize the graphs with the property that PD is a complete intersection for all orientations of G. It has been shown that ring graphs and complete graphs have this property [5]. Acknowledgements I would like to thank Rafael H. Villarreal and Isidoro Gitler for their very valuable comments and suggestions to improve this work. Enrique Reyes Departamento de Matem´ aticas, CINVESTAV-IPN Apartado Postal 14-740, 07000 México City. [email protected]

References [1] Godsil C.; Royle G., Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer, 2001. [2] Harary F., Graph Theory, Addison-Wesley, Reading, MA, 1972. [3] Ishizeki T., Analysis of Grobner bases for toric ideals of acyclic tournament graphs, Masters thesis, The University of Tokyo, 2000. [4] Oxley J., Matroid Theory, Oxford University Press, Oxford, 1992.

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[5] Reyes E., Toric ideals and affine varieties, blowup algebras and combinatorial optimization problems, PhD thesis, Cinvestav–IPN, 2006. [6] Schrijver A., Theory of Linear and Integer Programming, John Wiley & Sons, New York, 1986. [7] Sturmfels B., Gröbner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, Rhode Island, 1996. [8] Villarreal R.H., Rees algebras of edge ideals, Comm. Algebra 23 (1995), 3513–3524.