Bounded degree conjecture holds precisely for c-crossing-critical graphs with c<=12
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | COMBINATORICA |
| MU Faculty or unit | |
| Citation | |
| web | |
| Doi | https://doi.org/10.1007/s00493-021-4285-3 |
| Keywords | Crossing number; Crossing-critical; Exhaustive generation; Path-width |
| Description | 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. |
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