On the universal Gröbner bases of toric ideals of graphs

Abstract

The universal Gröbner basis of an ideal is a Gröbner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.

Toric ideals and diagonal 2-minors

“…As Example 3.10 demonstrates, the equality U (P G ) = C(I G ⋆ ) does not hold in general. From Theorem 3.4 in [12] we have that a binomial B belongs to the universal Gröbner basis of P G if and only if B = B Γ , where Γ is a mixed even closed walk of G ⋆ . For the definition of a mixed walk see [12].…”

“…In general, it is not easy to describe the set U (I) or even to estimate the degree of its elements. The case that I is a toric ideal has been of particular interest, see for example [2] and [12].…”

“…Our approach produces new examples of non-bipartite graphs H such that the toric ideal I H has a quadratic Gröbner basis. Moreover, using Theorem 3.8 and Theorem 3.4 in [12], we characterize in graph theoretical terms the elements of the universal Gröbner basis of P G , under the assumption that G is a connected graph with at most one cycle.…”

Toric ideals associated with gap-free graphs

“…Complete characterizations of both C G (Villarreal [28]) and UGB(I G ) (Tatakis and Thoma [26]) are known. We recall the characterization of C G (using the phrasing in Ohsugi and Hibi's article [20]) as a reference.…”

“…Another strong connection between the realms of commutative algebra and combinatorics is the one which links initial ideals of the toric ideal I G to triangulations of the edge polytope of G, see Sturmfels's book [25] and the recent article by Haase, Paffenholz, Piechnik and Santos [6]. Furthermore, Gröbner bases of I G have been studied among others by Ohsugi and Hibi [21] and Tatakis and Thoma [26]. A necessary condition for I G to have a squarefree initial ideal is the normality of K [G], which was characterized combinatorially by Ohsugi and Hibi [19] and Simis, Vasconcelos and Villarreal [23].…”