arXiv:0806.0585v3 [math.AC] 28 Apr 2009 PROPERTIES OF CUT IDEALS ASSOCIATED TO RING GRAPHS UWE NAGEL * AND SONJA PETROVI ´ C Abstract. A cut ideal of a graph records the relations among the cuts of the graph. These toric ideals have been introduced by Sturmfels and Sullivant who also posed the problem of relating their properties to the combinatorial structure of the graph. We study the cut ideals of the family of ring graphs, which includes trees and cycles. We show that they have quadratic Gr¨ obner bases and that their coordinate rings are Koszul, Hilbertian, and Cohen-Macaulay, but not Gorenstein in general. 1. Introduction Let G be any finite (simple) graph with vertex set V (G) and edge set E(G). In [16] Sturmfels and Sullivant associate a projective variety X G to G as follows. Let A|B be an unordered partition of the vertex set of G. Each such partition defines a cut of the graph, denoted by Cut(A|B), which is the set of edges {i,j } such that i ∈ A, j ∈ B or j ∈ A, i ∈ B. For each A|B, we can then assign variables to the edges according to whether they are in Cut(A|B) or not. The coordinates q A|B are indexed by the unordered partitions A|B, and the variables encoding whether the edge {i,j } is in the cut are s ij and t ij (for ”separated” and ”together”). The variety X G , which we call the cut variety of G, is specified by the following homomorphism between polynomial rings: φ G : K [q A|B : A|B partition] → K [s ij ,t ij : {i,j } edge of G], q A|B →  {i,j }∈Cut(A|B) s ij  {i,j }∈E(G)\Cut(A|B) t ij The cut ideal I G is the kernel of the map φ G . It is a homogeneous toric ideal (note that deg φ G (q A|B )= |E(G)|). The variety X G is defined by the cut ideal I G . Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables. However, the algebraic properties of cut ideals are largely unknown. It is clear that the properties of the cut ideal depend on the combinatorics of the graph. Sturmfels and Sullivant pose the following conjecture. Conjecture 1.1 ([16], Conjecture 3.7.). The semigroup algebra K [q ]/I G is normal if and only if K [q ]/I G is Cohen-Macaulay if and only if G is free of K 5 minors. We provide evidence for this conjecture by establishing it for a large class of such graphs. We refer to Section 6 for the definition of ring graphs. Theorem 1.2. The cut ideal of a ring graph admits a squarefree quadratic Gr¨ obner basis. Hence, its coordinate ring is Cohen-Macaulay. This and further results on cut ideals of ring graphs are established in Section 6. The next section is a collection of necessary definitions and results that we use repeat- edly. In particular, we recall the theorem of Sturmfels and Sullivant [16] about clique * The work for this paper was done while the first author was sponsored by the National Security Agency under Grant Number H98230-07-1-0065. 1