Which Cubic Graphs have Quadrangulated Spherical Immersions?

TL;DR

This paper classifies and constructs spherical quadrangulations with vertices of degree 3 or 4, linking them to cubic graph immersions, and provides methods and computational results for these embeddings.

Contribution

It introduces four new methods for constructing spherical quadrangulations and classifies disk quadrangulations with degree constraints, connecting them to cubic graph immersions.

Findings

Classified all quadrangulations of the disk with degree constraints.

Developed four construction methods for spherical quadrangulations.

Computed embeddings for nearly all cubic multigraphs on eight vertices.

Abstract

We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a crossing, these embeddings can also be thought of as transversal immersions of cubic graphs which we refer to as the {\it extracted graphs}. We also consider quadrangulations of the disk in which interior vertices have degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all such quadrangulations of the disk. Then, we provide four methods for constructing spherical quadrangulations, two of which use quadrangulations of the disk as input. Two of these methods provide one-parameter families of quadrangulations, for which we prove that the sequence of isomorphism types of extracted graphs is periodic. We close with a description of computer computations which yielded spherical quadrangulations for all but three cubic multigraphs on eight vertices.