Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12
Drago Bokal, Zdenĕk Dvořák, Petr Hliněný, Leaños, Jesús, Bojan Mohar, Tilo Wiedera · arXiv (Cornell University) · 2019
We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For every fixed pair of integers with $c\ge 13$ and $d\ge 1$, we give first explicit constructions of $c$-crossing-critical graphs containing a vertex of degree greater than $d$. We also show that such unbounded degree constructions do not exist for $c\le 12$, precisely, that there exists a constant $D$ such that every $c$-crossing-critical graph with $c\le 12$ has maximum degree at most $D$. Hence, the bounded maximum degree conjecture of $c$-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values $c\le 12.$