[Paper Review] Simple polytopes arising from finite graphs

Hidefumi Ohsugi, Takayuki Hibi|arXiv (Cornell University)|Apr 27, 2008

Commutative Algebra and Its Applications4 references3 citations

TL;DR

This paper classifies finite graphs whose edge polytopes are simple, proves that such polytopes have toric ideals with quadratic Gröbner bases, and computes their Ehrhart polynomials and normalized volumes. The key contribution is a complete characterization of graphs yielding simple edge polytopes, linking combinatorial graph structure to algebraic and geometric properties of associated polytopes.

ABSTRACT

Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Gröbner basis if the edge polytope ${\cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.

Motivation & Objective

• To classify all finite graphs whose edge polytope is simple.
• To establish the existence of a quadratic Gröbner basis for the toric ideal of a simple edge polytope.
• To compute the Ehrhart polynomial and normalized volume of simple edge polytopes.
• To clarify the relationship between simplicity, smoothness, and the absence of simplicial structure in edge polytopes.

Proposed method

• Define the edge polytope as the convex hull of vectors ρ(e) associated with edges and loops in a finite graph G.
• Use the condition (*) that ensures no edge is redundant when both endpoints have loops.
• Characterize simplicity of the edge polytope via combinatorial conditions on the graph, particularly involving induced subgraphs and degrees.
• Apply Gröbner basis theory to show that the toric ideal IG admits a quadratic, squarefree initial ideal when PG is simple.
• Use the Hilbert function of the edge ring K[G] to compute the Ehrhart polynomial, leveraging the existence of a squarefree initial ideal.
• Derive explicit formulas for the Ehrhart polynomial and normalized volume based on graph structure: complete bipartite graphs, graphs with loops, and induced subgraphs.

Experimental results

Research questions

• RQ1 Which finite graphs G yield a simple edge polytope PG?
• RQ2 Does the toric ideal IG of a simple edge polytope always possess a quadratic Gröbner basis?
• RQ3 When is a simple edge polytope also smooth, and how does this relate to its combinatorial structure?
• RQ4 What is the Ehrhart polynomial of a simple edge polytope in terms of graph parameters?
• RQ5 How can the normalized volume of a simple edge polytope be computed from the graph?

Key findings

• A finite graph G has a simple edge polytope PG if and only if it satisfies specific structural conditions: no edge is redundant when both endpoints have loops, and certain induced subgraphs have no vertices of degree 1 or no even cycles of a certain form.
• For any graph G whose edge polytope PG is simple, the toric ideal IG admits a quadratic Gröbner basis with squarefree initial monomials.
• The edge polytope PG is simple but not a simplex if and only if it is smooth but not a simplex.
• For a complete bipartite graph G with parts of size p and q, the Ehrhart polynomial is i(PG, m) = (p+m-1 choose p-1)(q+m-1 choose q-1), and the normalized volume is (p+q-2 choose p-1).
• If G has p loops and d vertices, with W the set of non-loop vertices, then i(PG, m) = sum_{j=1}^p (j+m-2 choose j-1)(d-j+m choose d-j), and the normalized volume is sum_{j=1}^p (d-1 choose j-1).
• The Ehrhart polynomial of PG coincides with the Hilbert function of the edge ring K[G], which is normal due to the existence of a squarefree initial ideal.