Chip-firing on the Platonic solids: a primer for studying graph gonality

TL;DR

This paper introduces chip-firing games and graph gonality using Platonic solid graphs, demonstrating techniques and proving gonality for dodecahedron and icosahedron graphs for the first time.

Contribution

It provides the first proofs of gonality for the dodecahedron and icosahedron graphs, expanding understanding of graph gonality through classical geometric graphs.

Findings

Dodecahedron graph gonality is 6.

Icosahedron graph gonality is 9.

Illustrates tools like independent sets, treewidth, and Dhar's algorithm.

Abstract

This paper provides a friendly introduction to chip-firing games and graph gonality. We use graphs coming from the five Platonic solids to illustrate different tools and techniques for studying these games, including independent sets, treewidth, scramble number, and Dhar's burning algorithm. In addition to showcasing some previously known results, we present the first proofs that the dodecahedron graph has gonality $6$, and that the icosahedron graph has gonality~$9$.