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Bounded degree conjecture holds precisely for ▫$c$▫-crossing-critical graphs with ▫$c \leq 12$▫ [Elektronski vir]
Bokal, Drago, 1978- ...
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 ▫$c\le 12$▫.
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