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

Abstrakt

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 Dvorák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c <=12.

Klíčová slova

• Bounded
• maximum
• degree
• conjecture
• holds
• precisely
• for
• c-crossing-critical
• graphs
• with
• c<=12