Gunnar Brinkmann ; Thomas Tucker ; Nico Van Cleemput - On the genera of polyhedral embeddings of cubic graph

dmtcs:6729 - Discrete Mathematics & Theoretical Computer Science, November 5, 2021, vol. 23, no. 3 - https://doi.org/10.46298/dmtcs.6729
On the genera of polyhedral embeddings of cubic graph

Authors: Gunnar Brinkmann ; Thomas Tucker ; Nico Van Cleemput

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.


Volume: vol. 23, no. 3
Section: Graph Theory
Published on: November 5, 2021
Accepted on: October 6, 2021
Submitted on: August 24, 2020
Keywords: Mathematics - Combinatorics


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