On the genera of polyhedral embeddings of cubic graph

TL;DR

This paper investigates the existence and properties of polyhedral embeddings of cubic graphs, providing theoretical insights, an efficient algorithm, and computational results up to 28 vertices, revealing the rarity of such embeddings.

Contribution

It introduces a new algorithm for computing all polyhedral embeddings of cubic graphs and demonstrates that such embeddings can exist in arbitrarily high genus, with many cubic graphs lacking any polyhedral embedding.

Findings

Polyhedral embeddings can exist in arbitrarily high genus.

Most cubic graphs up to 28 vertices do not have polyhedral embeddings.

The ratio of cubic graphs without polyhedral embeddings increases with vertices.

Abstract

In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.