Publication details
On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees
| Authors | |
|---|---|
| Year of publication | 2025 |
| Type | Article in Periodical |
| Magazine / Source | DISCRETE MATHEMATICS |
| MU Faculty or unit | |
| Citation | |
| web | arXiv preprint |
| Doi | https://doi.org/10.1016/j.disc.2024.114347 |
| Keywords | Graph; Crossing number; Crossing-critical families |
| Description | A recent result of Bokal et al. (2022) [3] 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 that 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 provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to another long-standing question of this research area; we prove that for every c>=13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,...,d. |