Minimal generators of toric ideals of graphs

Abstract

Let I G be the toric ideal of a graph G. We characterize in graph theoretical terms primitive, minimal, indispensable and fundamental binomials of the toric ideal I G .

On the universal Gröbner bases of toric ideals of graphs

“…The Graver basis of a toric ideal of a graph has first been studied by H. Ohsugi and T. Hibi [4, Lemma 2.1] and the form of its elements was determined by E. Reyes, Ch. Tatakis and A. Thoma [6]. In [2, Theorem 5.1] J.…”

“…Tatakis and A. Thoma [6] describes the form of the primitive binomials, i.e. the elements B w ∈ I G that belong to the Graver basis.…”

“…since the cut vertices are counted twice, see Theorem 3.1 of [6]. Two times the degree of B w is the sum of edges of the cyclic blocks t 1 + · · · + t s 0 plus two times the number of cut edges s 1 , since cut edges are double edges of the walk w and edges of cycles are always single.…”

“…The knowledge of the form of the circuits [10,Proposition 4.2], the elements of the Graver basis [6], the minimal systems of generators [6] and the elements of the universal Gröbner basis of the toric ideal of a graph G, Theorem 3.4, allow us to produce examples of toric ideals having specific properties. For example, one can easily construct graphs such that the universal Gröbner basis is equal to the Graver basis, either by avoiding creating pure blocks in the elements of the Graver basis, or making subdivisions in some of the edges of pure blocks.…”

“…Thus, B w is an element of the Graver basis of I G , see [6], and has degree s(l + 1)/2. In the graph G there are a lot of circuits.…”