On the genera of polyhedral embeddings of cubic graph
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Publication:5024676
DOI10.46298/DMTCS.6729zbMATH Open1481.05111arXiv2008.09467OpenAlexW3212647122WikidataQ122602913 ScholiaQ122602913MaRDI QIDQ5024676
Author name not available (Why is that?)
Publication date: 27 January 2022
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/2008.09467
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