Minimal generators of toric ideals of graphs

On the universal Gröbner bases of toric ideals of graphs

“…since the cut points are counted twice, see Theorem 3.1 of [4]. Two times the degree of B w is the sum of edges of the cyclic blocks t 1 +• • •+t s0 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.…”

“…Circuits are always in the universal Gröbner basis [6] therefore the result follows. Theorem 3.1 and Corollary 3.2 of [4] imply that a primitive walk consists of blocks which are cut edges and cyclic blocks, one if it is a cycle otherwise at least two. Let w be a primitive walk and suppose that w has s 0 cyclic blocks and s 1 cut edges.…”

“…Thus s = s 0 + s 1 is the total number of blocks. From Theorem 3.1 of [4] we know that there are exactly s − 1 cut points and each one belongs to exactly two blocks. Let B 1 , .…”

“…Since any graph G with m vertices is a subgraph of the complete graph K m we have the following corollary. 4.13] and the elements of the universal Gröbner basis of the toric ideal of a graph G, Theorem 3.4 and the variety and the easyness of description of graphs permit us easily 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, just by avoiding creating pure blocks in the elements of the Graver basis or making subdivisions in some of the edges of pure blocks.…”

“…Let w be the walk that pass from every edge of the graph G. The length of the walk w is ls + s = s(l + 1), which is even. Then B w is an element of the Graver basis of I G , see [4], and has degree s(l + 1)/2. In the graph of G there are a lot of circuits and the degree of the binomial corresponding to the longest one, which consists of two odd cycles joined by a path of length s − 1, is (2l + 2(s − = l + s − 1.…”