On the genera of polyhedral embeddings of cubic graph

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.

Polyhedral Embeddings of Snarks with Arbitrary Nonorientable Genera

Mohar and Vodopivec [Combinatorics, Probability and Computing (2006) 15, 877-893] proved that for every integer $k$ ($k \geq 1$ and $k\neq 2$), there exists a snark which polyhedrally embeds in $\mathbb{N}_k$ and presented the problem: Is there a snark that has a polyhedral embedding in the Klein bottle? In the paper, we give a positive solution of the problem and strengthen Mohar and Vodopivec's result. We prove that for every integer $k$ ($k\geq 2$), there exists an infinite family of snarks with nonorientable genus $k$ which polyhedrally embed in $\mathbb{N}_k$. Furthermore, for every integer $k$ ($k> 0$), there exists a snark with nonorientable genus $k$ which polyhedrally embeds in $\mathbb{N}_k$.

ACHIRALITY AND PLANARITY

An embedding of a space Y into the 3-sphere S3 is said to be strictly achiral if its image is pointwise fixed by an orientation reversing homeomorphism of S3. A space Y is said to be abstractly planar if it can be embedded into the 2-sphere S2. We first extend Kuratowski Theorem into a criterion for abstract planarity of polyhedra, then show that a polyhedron has a strictly achiral polyhedral embedding into S3 if and only if it is abstractly planar. Some related higher dimensional examples are also discussed.