Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon

TL;DR

This paper proves the scissors congruence conjecture for a class of polytopes associated with cubic graphs, linking graph transformations to unimodular geometry and Ehrhart quasi-polynomials.

Contribution

It establishes the conjecture for polytopes related to cubic graphs using graph interchange operations and unimodular transformations.

Findings

Proof of the scissors congruence conjecture for cubic graph polytopes

Connection between graph operations and unimodular transformations

Characterization of Ehrhart quasi-polynomials for these polytopes

Abstract

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.