  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 Dvo\v{r}\'ak and Mohar
(without an explicit construction), holds true, surprisingly, exactly for the
values $c\le 12.$
