On complete intersection toric ideals of graphs

Abstract

We characterize the graphs G for which their toric ideals I G are complete intersections. In particular we prove that for a connected graph G such that I G is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.

Graphs and complete intersection toric ideals

“…This was first realized by Katzman [17, Remark 3.9], who provided a Möbius band with 8 vertices as an example of a complete intersection non planar graph. Later Tatakis and Thoma [28] provided another example which is not a Möbius band.…”

“…, c r 3 , c 1 ) three vertex disjoint odd primitive cycles of G ′ . By [28,Theorem 4.2], G ′ has either one or two non bipartite blocks and the proof falls naturally in two cases.…”

“…We need the following technical result which is included in the proof of [28,Theorem 5.3]. Recall that a block is a maximal connected subgraph B of G such that if one removes any of its vertices it is still connected.…”

“…Since ring graphs are obviously planar, they could derive that every complete intersection bipartite graph is planar, which was previously proved by Katzman [17] without using the notion of ring graph. When graphs are not necessarily bipartite there is some recent work by Tatakis and Thoma [28], in the last section we make use of some of their technical results. For directed graphs, the complete intersection property has also been widely studied, see for example [9,11,22].…”

Matroid Toric Ideals: Complete Intersection, Minors, and Minimal Systems of Generators

“…Since then, they have been extensively studied by several authors. In the context of toric ideals associated to combinatorial structures, the complete intersection property has been widely studied for graphs, see, e.g., [2,22,10]. In this work we address this problem in the context of toric ideals of matroids and prove that there are essentially three matroids whose corresponding toric ideal is a complete intersection; namely, the rank 2 matroids without loops or coloops on a ground set of 4 elements.…”