Date of Award

5-11-2025

Date Published

June 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Lee Kennard

Keywords

Topological graph theory;Toroidal embeddings;Torus obstructions

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

The embeddability of graphs into different surfaces has been studied for nearly a century. In 1930, Kuratowski famously proved that a graph is planar if and only if it does not contain the complete graph K5 or the complete bipartite graph K3,3 as a topological minor. In particular, K3,3 is the only cubic obstructions for the plane. This sparked interest in two different directions: What are analogous results for other surfaces, and what can be said about embeddings of K5 and K3,3 into other surfaces? Regarding the first question, it was shown in the 1970s and 80s by Glover, Huneke, Wang, and Archdeacon that a graph is projective planar if and only if it does not contain one of 103 graphs as a topological minor. For other surfaces, such as the torus, there are partial results at best, many of which have been obtained using computer algorithms. Regarding the embeddability of K5 and K3,3 into other surfaces, Mohar classified their embeddings into the real projective plane, while Gagarin, Kocay, and Neilson classified their embeddings into the torus. The main objective of this dissertation is to obtain a theoretical proof that the set of cubic torus obstructions of Betti number at most eight consists of ten graphs, and to specify these graphs. Our second main result is a classification of toroidal embeddings of the six cubic projective plane obstructions.

Access

Open Access

Download

Included in

Mathematics Commons

Share

COinS