Bounded maximum degree conjecture holds precisely for $c$-crossing-critical graphs with $c \leq 12$
Computational Geometry
2021-05-06 v2 Discrete Mathematics
Abstract
We study -crossing-critical graphs, which are the minimal graphs that require at least edge-crossings when drawn in the plane. For every fixed pair of integers with and , we give first explicit constructions of -crossing-critical graphs containing a vertex of degree greater than . We also show that such unbounded degree constructions do not exist for , precisely, that there exists a constant such that every -crossing-critical graph with has maximum degree at most . Hence, the bounded maximum degree conjecture of -crossing-critical graphs, which was generally disproved in 2010 by Dvo\v{r}\'ak and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values
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Cite
@article{arxiv.1903.05363,
title = {Bounded maximum degree conjecture holds precisely for $c$-crossing-critical graphs with $c \leq 12$},
author = {Drago Bokal and Zdeněk Dvořák and Petr Hliněný and Jesús Leaños and Bojan Mohar and Tilo Wiedera},
journal= {arXiv preprint arXiv:1903.05363},
year = {2021}
}