Abstract
We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find drawings with few crossings for each remaining graph. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal.
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Acknowledgements
We are greatly indebted to Tilo Wiedera and Markus Chimani for their tireless patience in manually running their exact crossing minimisation solver on many of the more difficult instances in this paper. We also thank Eric Weisstein for numerous useful conversations that significantly improved this paper. Finally, we thank the anonymous referee for their thoughtful comments and suggestions that significantly improved this paper.
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Clancy, K., Haythorpe, M., Newcombe, A. et al. There Are No Cubic Graphs on 26 Vertices with Crossing Number 10 or 11. Graphs and Combinatorics 36, 1713–1721 (2020). https://doi.org/10.1007/s00373-020-02204-6
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DOI: https://doi.org/10.1007/s00373-020-02204-6