Graphs and complete intersection toric ideals

Abstract

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal P G is a complete intersection or not. Whenever P G is a complete intersection, the algorithm also returns a minimal set of generators of P G . Moreover, we prove that if G is a connected graph and P G is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

Generalized multiplicities of edge ideals

“…Then the convex hull pyr a (F) of a and F is the pyramid over F with apex a and lattice height h(F) − u, a . Therefore we obtain (2) Vol n (pyr a (F)) = (h(F) − u, a )Vol n−1 (F).…”

“…, s. In particular, we recover Corollary 8.6 without using Proposition 5.3 and the volumes. For a study of simple graphs with complete intersection toric edge ideals, see [2,15,38].…”

Toric ideals and diagonal 2-minors

“…Since the equality P C = I C ⋆ holds, we deduce from Proposition 2.2 that the toric ideal I C ⋆ is complete intersection. Thus, from Proposition 6.1 in [1], the graph C ⋆ has exactly two vertex disjoint odd cycles, namely C and C ′ . By Lemma 3.2 in [7], every primitive binomial B ∈ I C ⋆ is of the form B = B Γ , where Γ is one of the following even closed walks: (1) Γ is an even cycle of C ⋆ , (2) Γ consists of two odd cycles of C ⋆ intersecting in exactly one vertex, (3) Γ = (C, {p i , q i }, C ′ ) where 1 ≤ i ≤ k, i.e.…”

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

“…We then give a complete list of all matroids whose corresponding toric ideal is a complete intersection (Theorem 2.3). To this end, we first give such a list for matroids of rank 2 (Proposition 2.2), which is based on results given in [2]. In Section 3, we provide a necessary condition for a matroid to contain a minor isomorphic to U d,2d for d ≥ 2 in terms of the values ∆ {B 1 ,B 2 } for B 1 , B 2 ∈ B (Proposition 3.3).…”

“…In this case, we associate to M the graph H M with vertex set E and edge set B. It turns out that I M coincides with the toric ideal of the graph H M (see, e.g., [2]). In particular, from [2, Corollary 3.9], we have that whenever I M is a complete intersection, then H M does not contain K 2,3 as subgraph, where K 2,3 denotes the complete bipartite graph with partitions of sizes 2 and 3.…”