Abstract
A surprising result of Bokal et al. proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is \(c=13\). The key to the result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We now provide a relatively short self-contained computer-free proof.
Supported by research grant GAČR 20-04567S of the Czech Science Foundation.
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References
Bokal, D., Dvořák, Z., Hliněný, P., Leaños, J., Mohar, B., Wiedera, T.: Bounded degree conjecture holds precisely for c-crossing-critical graphs with c \(\le \) 12. In: Symposium on Computational Geometry. LIPIcs, vol. 129, pp. 14:1–14:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). Full and improved version on arXiv:1903.05363
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Hliněný, P., Korbela, M. (2021). On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_9
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DOI: https://doi.org/10.1007/978-3-030-83823-2_9
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