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URN: urn:nbn:de:0030-drops-104183
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Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12

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Abstract

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.

BibTeX - Entry

@InProceedings{bokal_et_al:LIPIcs:2019:10418,
  author =	{Drago Bokal and Zdenek Dvor{\'a}k and Petr Hlinen{\'y} and Jes{\'u}s Lea{\~n}os and Bojan Mohar and Tilo Wiedera},
  title =	{{Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10418},
  URN =		{urn:nbn:de:0030-drops-104183},
  doi =		{10.4230/LIPIcs.SoCG.2019.14},
  annote =	{Keywords: graph drawing, crossing number, crossing-critical, zip product}
}

Keywords: graph drawing, crossing number, crossing-critical, zip product
Seminar: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue date: 2019
Date of publication: 2019


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