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We study c-crossing-critical graphs, which are the minimal graphs that require at least c edgecrossings 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. 2012 ACM Subject Classification Theory of computation→ Computational geometry; Mathematics of computing → Graphs and surfaces

Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12.