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

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 quasipolynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set {0, 1 4 , 1 2 , 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 quasipolynomial 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.

Rational Ehrhart Theory

“…Example. The recent paper [10] studied certain families of polytopes arising from graphs, which exhibit period collapse. One example is the pyramid P 5 := conv (0, 0, 0) , 1 2 , 0, 0 , 0, 1 2 , 0 , 1 2 , 1 2 , 0 , 1 4 , 1 4 , 1…”