Bounded maximum degree conjecture holds precisely for c-crossing-critical graphs with c ≤ 12

03/13/2019 ∙ by Drago Bokal, et al. ∙ 0

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 a vertex 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.

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