Abstract
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 >= 13 and d >= 1, we give first explicit constructions of c-crossing-critical graphs containing arbitrarily many vertices of degree greater than d.
We also show that such unbounded degree constructions do not exist for c <= 12, precisely, that there exists a constant D such that every c-crossing-critical graph with c <= 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 <= 12.