The universal Gröbner basis of a binomial edge ideal

Transcription

1 The universal Gröbner basis of a binomial edge ideal Mourtadha Badiane Isaac Burke Emil Sköldberg National University of Ireland, Galway Galway, Ireland {m.badiane1,i.burke1,emil.skoldberg}@nuigalway.ie Submitted: Feb 3, 2016; Accepted: Sep 28, 2017; Published: Oct 6, 2017 Mathematics Subject Classifications: 13P10, 05E40 Abstract We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph. Keywords: Binomial edge ideals; Parity binomial edge ideals; universal Gröbner basis; Graver basis 1 Introduction For n N >0, [n] := {1,..., n}. Let G be a simple graph on the vertex set [n], that is, G has no loops and no multiple edges. Let E(G) denote the edge set of G. Let F be a field and let S = F [x 1,..., x n, y 1,..., y n ] be the polynomial ring in 2n variables. The binomial edge ideal of G was introduced and studied independently by Herzog, Hibi, Hreinsdóttir, Kahle and Rauh [2] and Ohtani [6]. Definition 1.1. The binomial edge ideal of G is J G := x i y j x j y i : {i, j} E(G) S. (1.1) The parity binomial edge ideal of G was introduced and studied by Kahle, Sarmiento and Windisch [5] but had previously been examined by Herzog, Macchia, Madani and Welker [3]. Supported by the Irish Research Council and the Hardiman Scholarship Scheme the electronic journal of combinatorics 24(4) (2017), #P4.11 1

2 Definition 1.2. The parity binomial edge ideal of G is I G := x i x j y i y j : {i, j} E(G) S. (1.2) These ideals appear in various settings and applications in mathematics and statistics and belong to an important class of binomial ideals which may be defined as follows. If we let R = F [x 1,..., x n ] then an ideal I of R is a pure difference ideal (also known in the literature as a pure binomial ideal) if I is generated by differences of monic monomials i.e. binomials of the form x u x v with u, v N n. There are several well-known distinguished subsets of binomials in such an ideal I, two of which we now mention. A binomial x u x v I is called primitive if there exists no other binomial x u x v I such that x u divides x u and x v divides x v. The set of primitive binomials in I is called the Graver basis of I and denoted Gr(I). The union of all of the reduced Gröbner bases of I is called the universal Gröbner basis of I and denoted U(I). Graver bases were originally defined for toric ideals by Sturmfels [8]. Charalambous, Thoma and Vladoiu [1] recently generalised the concept to an arbitrary pure difference ideal I. They show that Gr(I) is finite and extend an argument of Sturmfels to prove that U(I) Gr(I). One open problem that arises in the literature is providing a combinatorial characterisation of toric ideals for which the universal Gröbner basis and the Graver basis are equal (many examples have been discovered, see Petrović, Thoma and Vladoiu [7] and references therein). We consider this problem for certain classes of pure difference ideals that are not lattice ideals. In particular, we show that U(J G ) = Gr(J G ) and provide a description for this basis set in terms of certain paths in G. We conjecture a similar result for I G and prove this conjecture for the case when G is the complete graph. 1.1 Preliminaries Throughout the paper we assume that G is finite, undirected and connected. For any W [n], let G[W ] denote the induced subgraph on W, and for a sequence of vertices π = (i 0,..., i r ) [n] r+1, G[π] := G[{i 0,..., i r }]. A (v, w)-path of length r is a sequence of vertices v = i 0, i 1,..., i r = w such that {i k, i k+1 } E(G) for all k = 0,..., r 1. The path is odd (even) if its length is odd (even). The interior of a (v, w)-path π = (i 0,..., i r ) is the set int(π) = {i 0,..., i r } \ {v, w}. The inverse π 1 of a (v, w)-path π = (i 0,..., i r ) is the (w, v)-path (i r, i r 1,..., i 0 ). For the vertex set of a graph H we sometimes use the notation V (H). For a monomial x u = x d 1 1 y e 1 1 x dn n yn en in S the set {i : d i 0 or e i 0} [n] is denoted by V (x u ). 2 Binomial Edge Ideals In this section we will use two different gradings on S, the first is the N 2 -grading by considering the letter of a variable, so we let ldeg(x i ) = (1, 0) and ldeg(y i ) = (0, 1) for all i [n]. The second is the N n -grading which considers the vertex of a variable and we set gdeg(x i ) = gdeg(y i ) = e i for all i [n], where e i is the ith standard basis vector in N n. The the electronic journal of combinatorics 24(4) (2017), #P4.11 2

3 ideal J G is homogeneous with respect to both of these gradings and we combine them into what we call the multidegree of a monomial mdeg(x u ) := (ldeg(x u ), gdeg(x u )) N 2 N n. We now recall the definition of admissible paths and the description of the Gröbner basis of J G with respect to the lexicographic order which was independently obtained by Herzog et al. [2] and Ohtani [6]. Definition 2.1. Fix a permutation σ S n of [n] and let i, j [n] satisfy σ 1 (i) < σ 1 (j). An (i, j)-path π = (i 0,..., i r ) in G is called σ-admissible, if (i) i k i l if k l; (ii) j 0,..., j s is not a path from i to j for any proper subset {j 0,..., j s } of {i 0,..., i r }; (iii) for each k = 1,..., r 1, either σ 1 (i k ) < σ 1 (i) or σ 1 (i k ) > σ 1 (j). Given a σ-admissible (i, j)-path π = (i 0,..., i r ) in G, where σ 1 (i) < σ 1 (j), u π := x ik. σ 1 (i k )<σ 1 (i) y ik σ 1 (i k )>σ 1 (j) Theorem 2.2 ([2], [6]). The set of binomials G G,σ := {u π (x i y j x j y i ) : π is a σ-admissible (i, j)-path in G} σ 1 (i)<σ 1 (j) is the reduced Gröbner basis of J G w.r.t. the lexicographic monomial order on S induced by x σ(1) x σ(n) y σ(1) y σ(n). Our first result is a characterisation of the binomials in J G. For this we need to introduce the following notations. We denote by d G (v, w) the length of a shortest (v, w)- path in G. For a monomial x u = x d 1 1 y e 1 1 x dn n yn en S, we sometimes use the notation deg xi (x u ) for d i and deg yi (x u ) for e i. For an induced subgraph H of G, we define the restriction of x u to H to be res H (x u ) = Π i V (H) x d i i ye i i. Lemma 2.3. Let x u x v be a multi-homogeneous binomial such that G[V (x u )] is a connected graph. Then x u x v J G. Proof. Suppose that m = x u is a monomial such that there is a pair of indices i < j with deg xi (m) 1 and deg yj (m) 1. Let G = G[V (m)]. We can assume that i and j are chosen such that d G (i, j) is minimal. Now we consider an (i, j)-path π = (i 0,..., i r ) of minimal length in G. Since π is of minimal length we can conclude that i k i l for k l and that no proper subset {j 0,..., j s } of {i 0,..., i r } is a path from i to j. Suppose that there is a k such that i < i k < j, then either deg xik (m) 1, in which case d G (i k, j) < d G (i, j) which contradicts the minimality of d G (i, j), or deg yik (m) 1, in which case d G (i, i k ) < d G (i, j) which again contradicts the minimality of d G (i, j). We may thus conclude that π is a σ-admissible (i, j)-path in G, where σ = id, the identity permutation in S n. the electronic journal of combinatorics 24(4) (2017), #P4.11 3

4 Now we consider the vertex i k on π. By the minimality of d G (i, j), if i k < i then deg xik (m) = 0 and if i k > j then deg yik (m) = 0. We may thus conclude that u π x i y j divides m, and therefore m is reducible with respect to G G,id. This shows that an irreducible monomial of the same multidegree as x u has the form x w = y e 1 i 1 y e k i k x d k i k x d l i l where i 1 < i 2 < < i l. Since there is only one such monomial in a given multidegree, we can conclude that x u x v reduces to zero with respect to G G,id and thus x u x v J G. Lemma 2.4. A multi-homogeneous binomial x u x v lies in J G if and only if mdeg(res C (x u )) = mdeg(res C (x v )) for all connected components C of G[V (x u )]. Proof. Let x u x v J G, then we can write x u x v = k x w k (x ik y jk x jk y ik ) where {i k, j k } E(G) for all k. Let C be a component in G[V (x u )], then by restricting to C we get res C (x u ) res C (x v ) = res C (x w k )(x ik y jk x jk y ik ) k,i k V (C) since j k V (C) if and only if i k V (C). Thus we see that res C (x u ) res C (x v ) J G, and therefore mdeg(res C (x u )) = mdeg(res C (x v )). For the converse, suppose x u x v satisfies that mdeg(res C (x u )) = mdeg(res C (x v )) for all connected components C 1,... C r of G[V (x u )], then, by Lemma 2.3, res Ci (x u ) res Ci (x v ) J G for all r, and we can write x u x v = r i=1 x u i 1 j=1 res C i (x v ) i j=1 res C i (x u ) (res C i (x u ) res Ci (x v )), which shows that x u x v J G. Definition 2.5. An (i, j)-path π = (i 0,..., i r ) in G is called weakly admissible if it satisfies conditions (i) and (ii) of the definition of a σ-admissible path. Given a weakly admissible (i, j)-path π = (i 0,..., i r ) in G, S π := {t i1 t i2 t ir 1 (x i y j x j y i ) : t k {x k, y k }} and S(J G ) := π S π \ {0} where π runs over all weakly admissible paths in G. Notice that if π is an (i, i)-path in G, then π is weakly admissible if and only if π is the path (i) of length 0, in which case S π = {0}. the electronic journal of combinatorics 24(4) (2017), #P4.11 4

5 Figure 1: See Examples 1 and 2. Example 1. Let G be the graph in Figure 1. The weakly admissible paths in G are the paths (1), (2), (3), (4), (1, 2), (1, 3), (2, 3), (2, 4), (1, 2, 4) and (3, 2, 4), together with their inverses. Hence S(J G ) = 16. Theorem 2.6. The sets S(J G ), U(J G ) and Gr(J G ) coincide. Proof. We prove the theorem in three steps; the containments S(J G ) U(J G ), U(J G ) Gr(J G ) and Gr(J G ) S(J G ). Step 1. S(J G ) U(J G ): Let π = (i 0,..., i r ) be a weakly admissible (i, j)-path in G and let f = t i1 t i2 t ir 1 (x i y j x j y i ) be a corresponding binomial in S(J G ). Now let σ S n be a permutation such that σ 1 (i) < σ 1 (j), σ 1 (i k ) < σ 1 (i) for all k such that t ik = y ik and σ 1 (i k ) > σ 1 (j) for all k such that t ik = x ik. Then f G G,σ and thus f U(J G ). Step 2. U(J G ) Gr(J G ): [1, Proposition 4.2]. Step 3. Gr(J G ) S(J G ): Let x u x v be a primitive binomial in J G and let C be a component of G[V (x u )]. By Lemma 2.4 we have that res C (x u ) res C (x v ) J G, so by primitivity we can conclude that C := G[V (x u )] is connected. Since u v we can choose i, j V (x u ) such that d C (i, j) is minimal among all pairs i, j with deg xi (x u ) > deg xi (x v ) and deg yj (x u ) > deg yj (x v ). Now let π = (i 0,..., i r ) be an (i, j)-path in C of minimal length. Suppose that there is a k {1,..., r 1} such that deg xik (x u ) deg xik (x v ). Then either deg xik (x u ) > deg xik (x v ), in which case d C (i k, j) would contradict the minimality of d C (i, j), or deg xik (x u ) < deg xik (x v ), in which case by homogeneity we would have deg yik (x u ) > deg yik (x v ) and thus d C (i, i k ) would contradict the minimality of d C (i, j). So for k {1,..., r 1} we have deg xik (x u ) = deg xik (x v ) and hence by homogeneity deg yik (x u ) = deg yik (x v ). We can then, for k = 1,..., r 1, let z ik {x ik, y ik } such that z ik divides x u and thus also x v. Then z i1 z ir 1 (x i y j x j y i ) J G with z i1 z ir 1 x i y j x u and z i1 z ir 1 x j y i x v, which implies that x u x v = z i1 z ir 1 (x i y j x j y i ) S(J G ). 3 Parity Binomial Edge Ideals In this section we will use two different gradings on S, but not exactly as in the previous section. The first grading is the Z 2 2-grading by considering the letter of a variable, so we let ldeg(x i ) = (1, 0) Z 2 2 and ldeg(y i ) = (0, 1) Z 2 2 for all i [n]. The second is the N n -grading as in the previous section. The ideal I G is homogeneous with respect to both of these gradings and we combine them into what we call the multidegree of a monomial mdeg(x u ) := (ldeg(x u ), gdeg(x u )) Z 2 2 N n. the electronic journal of combinatorics 24(4) (2017), #P4.11 5

6 Lemma 3.1 ([5]). Let π = (i 0,..., i r ) be an (i, j)-path in G and for k int(π), let t k {x k, y k } be arbitrary. If π is odd, then (x i x j y i y j ) t k I G. k int(π) If π is even, then (x i y j y i x j ) t k I G. k int(π) Definition 3.2. An (i, j)-path π in G is called minimal, if (i) for no k int(π) there is an (i, j)-path with the same parity as π in G[π \ {k}]; (ii) there is no shorter (i, j)-path π in G satisfying parity(π ) = parity(π) and int(π ) = int(π). For a minimal (i, j)-path π in G, we define a set of binomials S π as follows. If π is odd, then S π := S + π,o S π,o where S + π,o := {(x i x j y i y j ) S π,o := {(y i y j x i x j ) If π is even, then S π := S π,e where S π,e := {(x i y j y i x j ) k int(π) k int(π) k int(π) t k : t k {x k, y k }}, t k : t k {x k, y k }}. t k : t k {x k, y k }}. S(I G ) := π S π \ {0} where π runs over all minimal paths in G. Notice that if π is an even (i, i)-path in G, then π is minimal if and only if π is the path (i) of length 0, in which case S π = {0}. Example 2. Let G be the graph in Figure 1. The minimal paths in G are the paths (1), (2), (3), (4), (1, 2), (1, 3), (2, 3), (2, 4), (1, 2, 3), (1, 2, 4), (1, 3, 2), (2, 1, 3), (3, 2, 4), (1, 2, 3, 1), (1, 3, 2, 4), (2, 1, 3, 2), (3, 1, 2, 3), (3, 1, 2, 4), (2, 1, 3, 2, 4), (2, 3, 1, 2, 4) and (4, 2, 1, 3, 2, 4), together with their inverses. Hence S(I G ) = 92. the electronic journal of combinatorics 24(4) (2017), #P4.11 6

7 Given a graph G, it is clear that the set of its weakly admissible paths is a subset of the set of its minimal paths. If π is a minimal (i, j)-path in G which is not a weakly admissible path, π contains repeated vertices or G[π] contains an (i, j)-path π of parity opposite to that of π such that int(π ) int(π). Conjecture 3.3. The sets S(I G ), U(I G ) and Gr(I G ) coincide. We have partially tested Conjecture 3.3 for small graphs using the software gfan [4] and have found no counterexamples so far. It must be said, however, that we are not currently aware of any algorithm for computing the Graver basis of an arbitrary pure difference ideal. The main result of this section is a proof that Conjecture 3.3 holds when G is the complete graph K n on the vertex set [n]. The rest of the section is arranged as follows. In Lemmas 3.4 through 3.7 we describe a reduced Gröbner basis of I Kn. In Lemmas 3.8 through 3.12 we characterise the binomials in I Kn. In Theorem 3.13 the main result is proved. Lemma 3.4. The minimal paths in K n are all (i), (i, j), (i, k, j) and (i, k, l, i) where i, j, k and l are distinct elements of [n]. Proof. Let π = (i 0,..., i r ) be an (i, j)-path in K n. Suppose that π is odd. If int(π) = then π is necessarily of the form (i, j, i, j,..., i, j) and thus minimal if and only if π = (i, j). If int(π) then there are two cases: i = j and i j. If i = j then int(π) 2 so let i s1 i s2 int(π) and notice that K n contains the odd path π = (i 0, i s1, i s2, i 0 ). Now K n [π \ {i s1 }] = K n [π \ {i s2 }] = K 2 which does not contain an odd cycle, hence π is minimal. It follows that π is minimal if and only if π = (i, k, l, i), where {k, l} = int(π). If i j then for all k int(π) the graph K n [π \ {k}] contains the odd (i, j)-path π = (i, j), hence π is not minimal in this case. The proof for an even path is similar and omitted. Given a permutation σ S n of [n] and a set L [n] let denote the lexicographic monomial order on S induced by t σ(1) t σ(n) t σ(1) t σ(n) where t σ(i) = x σ(i), t σ(i) = y σ(i) for all i [n] \ L and t σ(i) = y σ(i), t σ(i) = x σ(i) for all i L. For i, j [n] write i j if σ 1 (i) < σ 1 (j). Let G (G) denote the reduced Gröbner basis of I G with respect to. For a nonzero f S let N G (G)(f) denote the normal form of f with respect to G (G) and let in (f) denote the initial monomial of f with respect to. For the next three lemmas (3.5 to 3.7) fix a permutation σ S n and a set L [n]. For v [n] c v := { +1 if σ 1 (v) / L 1 if σ 1 (v) L; r v := { y v if σ 1 (v) / L x v if σ 1 (v) L. the electronic journal of combinatorics 24(4) (2017), #P4.11 7

8 For i, j, k, l [n] let B (i,j) = {c i (x i x j y i y j ) : i j}, B (i,k,j) = {c i (x i y j y i x j )r k : i, k j} and B (i,k,l,i) = {c i (x 2 i yi 2 )r k r l : k, l i}. Notice that for an element f ( ) B (i,j) B (i,k,j) B (i,k,l,i), the value of ci ensures that the coefficient of the initial monomial in (f) is 1. Finally let the set Γ B (i,k,j) consist of all binomials f = c i (x i y j y i x j )r k B (i,k,j) satisfying i k j and {σ 1 (i), σ 1 (k)} L = 1. Lemma 3.5. Let f S be a nonzero binomial corresponding to a minimal path π in K n (in the sense of Lemma 3.1). Then f is reduced with respect to if and only if f Λ := ( B (i,j) B (i,k,j) B (i,k,l,i) ) \ Γ. Proof. By Lemma 3.4 it suffices to consider only binomials corresponding to the paths (i, j), (i, k, j) and (i, k, l, i) in K n, where i, j, k and l are distinct elements of [n]. Without loss of generality we may assume that i j. If f is the binomial corresponding to the path (i, j) then clearly f is reduced if and only if f B (i,j) Λ. If f is a binomial corresponding to the path (i, k, j) then there are three conceivable cases: i j k, i k j and k i j. For ease of notation f t (i,j,k) := c i(x i y k y i x k )t j where the superscript t {x, y} indicates whether t j = x j or t j = y j. Case 1 (i j k). If σ 1 (i) L then f(i,k,j) x is reduced by x ix k y i y k ; also f y (i,k,j) is reduced by f y (i,j,k) if σ 1 (j) L, or y j y k x j x k if σ 1 (j) L. If σ 1 (i) L then f(i,k,j) x is reduced by x j x k y j y k if σ 1 (j) L, or f(i,j,k) x if σ 1 (j) L; also f y (i,k,j) is reduced by y i y k x i x k. Case 2 (i k j). If σ 1 (i) L then f x (i,k,j) is reduced by x ix k y i y k ; also f y (i,k,j) is reduced by y k y j x k x j if σ 1 (k) L but is irreducible if σ 1 (k) L. If σ 1 (i) L then f x (i,k,j) is reduced by x kx j y k y j if σ 1 (k) L but is irreducible if σ 1 (k) L; also f y (i,k,j) is reduced by y i y k x i x k. Case 3 (k i j). If σ 1 (i) L then f x (i,k,j) is reduced by x kx i y k y i if σ 1 (k) L but is irreducible if σ 1 (k) L; also f y (i,k,j) is reduced by y ky j x k x j if σ 1 (k) L but is irreducible if σ 1 (k) L. If σ 1 (i) L then f x (i,k,j) is reduced by x kx j y k y j if σ 1 (k) L but is irreducible if σ 1 (k) L; also f y (i,k,j) is reduced by y ky i x k x i if σ 1 (k) L but is irreducible if σ 1 (k) L. Finally let f = c i (x 2 i y 2 i )t k t l be a binomial corresponding to the path (i, k, l, i). If i k then f is reduced by one of f x (i,l,k), f y (i,l,k) or c i(x i x k y i y k ). The case i l is similar. If k, l i then by arguing as before one finds that f is irreducible if and only if t s = y s whenever σ 1 (s) L and t s = x s whenever σ 1 (s) L. Lemma 3.6. Let π be an (i, j)-path in K n and for k int(π), let t k {x k, y k } be arbitrary. Then (x i x j y i y j )Π k int(π) t k if π is odd and (x i y j y i x j )Π k int(π) t k if π is even, reduce to zero modulo Λ. Proof. It suffices to restrict to a minimal path π (if π is not minimal, then its binomial is a multiple of the binomial for a shorter path). If π is minimal, then Lemma 3.5 gives the result. Lemma 3.7. G (K n ) = Λ. the electronic journal of combinatorics 24(4) (2017), #P4.11 8

9 Proof. The proof is by Buchberger s criterion and is similar to the proof of Theorem 3.6 in Kahle et al. [5]. Let g, g Λ be reduced binomials corresponding, respectively, to the odd path π = (i, k, l, i) in K n and the even path π = (u, v, w) in K n. Consider the case that {i, k, l} {u, v, w} = 3. Since k, l i and u, v w we may assume without loss of generality that i = w, k = u and l = v. Thus g = c i (x 2 i y 2 i )r k r l, g = c k (x k y i y k x i )r l. There are four possibilities for the constants c i, c k. By duality we need only consider (c i, c k ) = (1, 1) and (c i, c k ) = ( 1, 1). If (c i, c k ) = (1, 1) then r k = y k and spol(g, g ) = (x k y 3 i y k x 3 i )y k r l = (y k y 2 i ) g + (y k x i ) g + 0 and in (spol(g, g )) in ( y k y 2 i g ), in (y k x i g). If (c i, c k ) = ( 1, 1) then spol(g, g ) = (x k x i y k y i )x i y k r l which is a monomial multiple of the binomial corresponding to the path (k, i) in K n and thus reduces to zero by Lemma 3.6. In a similar fashion to the above all spol(g, g ) (where g, g Λ) reduce to zero with respect to Λ. Thus the set Λ fulfills Buchberger s criterion and hence is a Gröbner basis of I Kn. By Lemma 3.5 it follows that the elements of Λ are reduced with respect to. Lemma 3.8. Let f = x u x v S be multi-homogeneous with V (x u ) = {i, j}. If gcd(x u, x v ) = 1 then f {±(x p i xq j yp i yq j ), ±(xp i yq j yp i xq j ) : p, q 1, parity(p) = parity(q)}. Proof. By homogeneity f is necessarily of the form x u x v = x d i i ye i i xd j j ye j j x d i i ye i i xd j j ye j j satisfying d i + e i = d i + e i and d j + e j = d j + e j. By gcd(x u, x v ) = 1 it follows that if d i > 0 then d i = 0 and similarly for all exponents. But if d i > 0 then d i > 0 or e i > 0 i.e. e i > 0 (since d i = 0) which in turn implies e i = 0. If d j > 0 we get a similar result. If d j = 0 then by V (x u ) = {i, j} we have e j > 0. By inverting the argument we obtain x u x v {±(x d i i xd j j y e i i ye j j ), ±(xd i i ye j j y e i i xd j j )}. The result follows from the implications of homogeneity. Lemma 3.9. Let be the lexicographic monomial order on S corresponding to σ = id and L =. Let x u = x d 1 1 y e 1 1 x dn n yn en S where V (x u ) > 2. Let k = max{i : i V (x u )} and let γ = k i=1 d i. { N G (Kn)(x u y d 1+e 1 1 y d k+e k k, if γ is even ) = x k y d 1+e 1 1 y d k 1+e k 1 k 1 y d k+e k 1 (3.1) k, if γ is odd. the electronic journal of combinatorics 24(4) (2017), #P4.11 9

10 Proof. The forms in (3.1) are clearly irreducible. By Lemma 3.7 we have x i x j y i y j G (K n ) for all i j [n], so that x u can be reduced to x u = x ds l s y d 1+e 1 1 y d s 1+e s 1 s 1 y es+l s y d s+1+e s+1 s+1 y d k+e k k for some 1 s k where l Z 0, l d s. By the homogeneity of x i x j y i y j parity( k i=1 d i) = parity(d s l). If d s = l then we are done. Otherwise we consider the following two cases. If 1 s k 1 then by V (x u ) > 2 there exists v V (x u ) \ {s}, v k. Since v, s k, by Lemma 3.7 f = y v (x s y k y s x k ) G (K n ). Using f and x s x k y s y k G (K n ) in that order x u can be reduced to x u = x ds l 2 s y d 1+e 1 1 y d s 1+e s 1 s 1 y es+l+2 s y d s+1+e s+1 s+1 y d k+e k k. Repeated iteration of this step gives one of the forms in (3.1), depending on the parity of d s l. If s = k then by V (x u ) > 2 there exist v 1, v 2 V (x u ) \ {k}, v 1 v 2. Since v 1, v 2 k, by Lemma 3.7 f = y v1 y v2 (x 2 k y2 k ) G (K n ). Using f we can reduce x u to one of the forms in (3.1), depending on the parity of d s l. Lemma Let be the lexicographic monomial order on S corresponding to σ = id and L =. Let x u = x d i i ye i i xd j j ye j j S i.e. V (x u ) 2. Let q = min{d i, d j }. N G (K n)(x u ) = x d i q i y e i+q i x d j q j y e j+q j. (3.2) Proof. The monomial x u can be reduced to (3.2) by the binomial x i x j y i y j G (K n ) (Lemma 3.7). No element of the set {in (g) : g G (K n )} divides (3.2) hence (3.2) is irreducible. Lemma Let f = x u x v S be such that V (x u ) > 2. Then f I Kn if f is multi-homogeneous. if and only Proof. : This is clear. : Suppose f is multi-homogeneous in degree ((α 1, α 2 ), β) Z 2 2 N n. Let be the lexicographic monomial order on S corresponding to σ = id and L =. Let k = max{i : i V (x u )}. If α 1 = 0 then N G (K n)(x u ) = N G (K n)(x v ) is the first form in (3.1) and if α 1 = 1 then N G (K n)(x u ) = N G (K n)(x v ) is the second form in (3.1). Lemma Let f = x u x v S be such that V (x u ) 2. Then f I Kn if f is of the form x d i i ye i i xd j j ye j j x d i q i y e i+q i x d j q j y e j+q j where q Z. if and only Proof. : f = x u x v I N G (K n)(x u ) = N G (K n)(x v ) where is the lexicographic monomial order on S corresponding to σ = id and L =. Apply Lemma : Apply Lemma Theorem Conjecture 3.3 holds for G = K n. Proof. Let G = K n throughout. We prove the theorem in three steps; the containments S(I G ) U(I G ), U(I G ) Gr(I G ) and Gr(I G ) S(I G ). the electronic journal of combinatorics 24(4) (2017), #P

11 Step 1. S(I G ) U(I G ): Here we invoke Lemmas 3.4 and 3.7. If π is a path of the form (i, k, l, i) in G then let σ S n be a permutation such that k, l i. A suitable choice of L {σ 1 (i), σ 1 (k), σ 1 (l)} then provides that (x 2 i y 2 i )r k r l G (G) or (y 2 i x 2 i )r k r l G (G). The cases π = (i, k, j) and π = (i, j) are similar and omitted. Step 2. U(I G ) Gr(I G ): [1, Proposition 4.2]. Step 3. Gr(I G ) S(I G ): Let f = x u x v Gr(I G ). If V (x u ) 2 then by Lemma 3.12 we have f = x d i i ye i i xd j j ye j j x d i q i y e i+q i x d j q j y e j+q j where q Z. If q > 0 then necessarily f = x i x j y i y j S(I G ). If q < 0 then necessarily f = y i y j x i x j S(I G ). Now consider the case that V (x u ) > 2. First suppose that f = t k (x u x v ) where k V (x u ) and t k {x k, y k }. Now x u x v is multi-homogeneous and it must be that V (x u ) = 2 since otherwise by Lemma 3.11 x u x v I G, contradicting the primitivity of f. Write f = t k (x u x v ) = t k (x d i i ye i i xd j j ye j j x d i i ye i i xd j j ye j j ). (3.3) If gcd(x u, x v ) 1 then it follows from the previous argument that f = t k t l (x ds s ys es x d s s y e s s ) where {s, l} {i, j} and t l {x l, y l }. Now x ds s ys es x d s s y e s s is multi-homogeneous and nonzero. These criteria are minimally satisfied by (d s, e s ) {(2, 0), (0, 2)} i.e. f = t k t l (x 2 s ys) 2 S(I G ) or f = t k t l (ys 2 x 2 s) S(I G ). For d s + e s > 2 the primitivity of f is contradicted either by one of these binomials or by an element of the form ±(x i x j y i y j ) S(I G ). If instead in (3.3) gcd(x u, x v ) = 1 then by Lemma 3.8 and the primitivity of f we have f = ±t k (x i y j y i x j ) S(I G ). Suppose now that f = x u x v cannot be written as t k (x u x v ) where k V (x u ) and t k {x k, y k }. Since f is multi-homogeneous and V (x u ) > 2 we can assume that for some i, j V (x u ) either x i x j x u and y i y j x v or y i y j x u and x i x j x v i.e. in this case the primitivity of f is contradicted by an element of the form ±(x i x j y i y j ) S(I G ). Acknowledgements The authors would like to thank the anonymous referee for their careful reading and helpful suggestions. References [1] Hara Charalambous, Apostolos Thoma, and Marius Vladoiu, Binomial fibers and indispensable binomials, Journal of Symbolic Computation 74 (2016), [2] Jürgen Herzog, Takayuki Hibi, Freyja Hreinsdóttir, Thomas Kahle, and Johannes Rauh, Binomial edge ideals and conditional independence statements, Advances in Applied Mathematics 45 (2010), no. 3, [3] Jürgen Herzog, Antonio Macchia, Sara Saeedi Madani, and Volkmar Welker, On the ideal of orthogonal representations of a graph in R 2, Advances in Applied Mathematics 71 (2015), the electronic journal of combinatorics 24(4) (2017), #P

12 [4] Anders N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, Available at [5] Thomas Kahle, Camilo Sarmiento, and Tobias Windisch, Parity binomial edge ideals, Journal of Algebraic Combinatorics 44 (2016), no. 1, [6] Masahiro Ohtani, Graphs and ideals generated by some 2-minors, Communications in Algebra 39 (2011), no. 3, [7] Sonja Petrović, Apostolos Thoma, and Marius Vladoiu, Bouquet algebra of toric ideals, arxiv: v2 (2015). [8] Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, the electronic journal of combinatorics 24(4) (2017), #P