On the period collapse of a family of Ehrhart quasi-polynomials

TL;DR

This paper investigates the period behavior of Ehrhart quasi-polynomials associated with polytopes derived from $ ext{1,3}$-graphs, revealing conditions under which the period collapses to 1 or 2, and exploring related combinatorial and geometric properties.

Contribution

It establishes bounds on the period of Ehrhart quasi-polynomials for these polytopes based on graph structure, and uncovers new combinatorial and geometric insights with potential broader applications.

Findings

Period of Ehrhart quasi-polynomial is at most 2 for trees and cubic graphs.

Period is exactly 4 for other $ ext{1,3}$-graphs.

New combinatorial and geometric properties of the associated polytopes were discovered.

Abstract

A graph whose nodes have degree 1 or 3 is called a $\{1,3\}$-graph. Liu and Osserman associated a polytope to each $\{1,3\}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.