Edge Polytopes and Edge Rings

Abstract

The convex polytopes arising from finite graphs and their toric ideals have been studied by many authors. The present chapter is devoted to introducing the foundation on the topics. In Section 5.1, we summarize basic terminologies on finite graphs. A basic fact on bipartite graphs is proved. The edge polytope of a finite graph is introduced in Section 5.2. We study the dimension, the vertices, the edges, and the facets of edge polytopes. In Section 5.3, the edge ring of a finite graph and its toric ideal is discussed. One of the main results is a combinatorial characterization for the toric ideal of an edge ring to be generated by quadratic binomials (Theorem 5.14). The problem of the normality of edge polytopes is studied in Section 5.4. It turns out that the odd cycle condition in the classical graph theory characterizes the normality of an edge polytope. Furthermore, it is shown that an edge polytope is normal if and only if it possesses a unimodular covering (Theorem 5.20). Finally, in Section 5.5, Gröbner bases of toric ideals arising from bipartite graphs will be discussed. In particular, we show that the toric ideal of the edge ring of a bipartite graph is generated by quadratic binomials if and only if it possesses a quadratic Gröbner basis (Theorem 5.27).