Properties Of Cut Ideals Associated To Ring Graphs

A cut ideal of a graph records the relations among the cuts of the graph. These toricideals 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 Grobner bases and that their coordinate rings are Koszul, Hilbertian, and Cohen-Macaulay, but not Goren-stein in general.