The universal Gr\"obner basis of a binomial edge ideal

We show that the universal Gr\"obner 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.


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 [HHH + 10] and Ohtani [Oht11]. The parity binomial edge ideal of G was introduced and studied by Kahle, Sarmiento and Windisch [KSW16] but had previously been examined by Herzog, Macchia, Madani and Welker [HMMW15].
Definition 1.2. The parity binomial edge ideal of G is (1.2) I G := x i x j − y i y j : {i, j} ∈ E(G) ⊆ S.
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 [Stu96]. Charalambous, Thoma and Vladoiu [CTV16] recently generalised the concept to an arbitrary pure difference ideal I, showing in particular that Gr(I) is finite and includes U(I) as a subset.
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 et al. [PTV15] and references therein). We consider this problem for certain classes of pure difference ideals which 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.

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 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 .
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 y en n ∈ S, we sometimes use the notation deg x i (x u ) for d i and deg y i (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 ( Proof. Suppose that m = x u is a monomial such that there is a pair of indices i < j with deg x i (m) ≥ 1 and deg y j (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 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 . Now we consider the vertex i k on π. By the minimality of d G ′ (i, j), if i k < i then deg x i k (m) = 0 and if i k > j then deg y i k (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 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 For the converse, suppose for all r, and we can write Given a weakly admissible (i, j)-path π = (i 0 , . . . , i r ) in G, 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 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( Step Step Step

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 .
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 + 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) 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 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). We have partially tested Conjecture 3.3 for small graphs using the software gfan [Jen] 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. 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 s 1 = i s 2 ∈ int(π) and notice that K n contains the odd path π ′ = (i 0 , i s 1 , i s 2 , i 0 ). Now K n [π ′ \{i s 1 }] ∼ = K n [π ′ \{i s 2 }] ∼ = 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 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] i − y 2 i )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 c i 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: 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 t k ∈ {x k , y k } 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.
Proof. The proof is by Buchberger's criterion and is similar to the proof of Theorem 3.6 in Kahle et al. [KSW16]. Let g, g ′ ∈ Λ be reduced binomials corresponding, respectively, to the odd path π = (v 0 , v 1 , v 2 , v 0 ) in K n and the even path π ′ = (u 0 , u 1 , u 2 ) in K n with u 0 ≻ u 2 . We write which is a monomial multiple of the binomial corresponding to the odd path (v 0 , u 2 ) in K n . Thus spol(g, g ′ ) reduces to zero by Lemma 3.6. The subcase σ −1 (v 0 ) ∈ L is dual to this. In a similar fashion 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 ≻.
Proof. By homogeneity f is necessarily of the form 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 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 y en 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 · · · y d k +e k k for some 1 ≤ s ≤ k where l ∈ Z ≥0 , l ≤ d s . By the homogeneity of . If d s = l then we are done. Otherwise we consider the following two cases.
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 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 ( Using f we can reduce x u ′ to one of the forms in (3.1), depending on the parity of d s − l.
Lemma 3.10. Let ≻ be the lexicographic monomial order on S corresponding to σ = id and L = ∅. Let 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 3.11. Let f = x u − x v ∈ S be such that |V (x u )| > 2. Then f ∈ I Kn if and only if f is multi-homogeneous.
Theorem 3.13. 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 ).
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 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 = 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 s − y 2 s ) ∈ S(I G ) or f = t k t l (y 2 s − 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 ).