Arithmetical rank of toric ideals associated to graphs

Abstract

Abstract. Let I G ⊂ K[x 1 , . . . , x m ] be the toric ideal associated to a finite graph G. In this paper we study the binomial arithmetical rank and the Ghomogeneous arithmetical rank of I G in 2 cases:(1) G is bipartite, (2) I G is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the G-homogeneous arithmetical rank coincide with the minimal number of generators of I G .

On the binomial arithmetical rank of toric ideals

“…We prove that for the toric ideals of such graphs, the binomial arithmetical rank equals the minimal number of generators under a suitable assumption (see Theorem 3.10). This is a generalization of the aforementioned results of [2]. As an application, we show that if the complement of a graph G is weakly chordal, then the toric ideal of G enjoys the property that the equality bar(I A ) = µ(I A ) holds.…”

“…So x z is not indispensable, and therefore B w is not an indispensable binomial. Consequently w has at least one In [2], the binomial arithmetical rank of I G was studied in two cases, namely when G is bipartite or I G is generated by quadratic binomials. For a bipartite graph G we have, from [4,Theorem 3.2], that the toric ideal I G is minimally generated by all circuits B w , where w is an even cycle with no chord.…”

“…The problem under consideration was examined for such ideals in [2]. More precisely, the author reveals two cases in which the binomial arithmetical rank coincides with the minimal number of generators, namely when the graph is bipartite or the toric ideal is generated by quadratic binomials.…”

Minimal generators of toric ideals of graphs

“…In graph theory there are several monomial or binomial ideals associated to a graph, see [4,7,10,11,15,24,25,27,30,31,33], depending on the properties one wishes to study. One of them is the toric ideal of a graph which has been extensively studied over the last years, see [4,7,9,8,12,13,18,19,20,21,22,29,32,31].…”

Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

“…For unexplained terminology and notation on graph theory and toric ideals we refer to [3,6,7,18]. Some references for toric ideals associated to graphs (without orientation) are [5,6,8,9,13,15,16,17].…”