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DOI: 10.4230/LIPIcs.SoCG.2019.14

Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12

Authors: Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

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

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Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera. Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bokal_et_al:LIPIcs.SoCG.2019.14,
  author =	{Bokal, Drago and Dvo\v{r}\'{a}k, Zden\v{e}k and Hlin\v{e}n\'{y}, Petr and Lea\~{n}os, Jes\'{u}s and Mohar, Bojan and Wiedera, Tilo},
  title =	{{Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c \langle= 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 =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.14},
  URN =		{urn:nbn:de:0030-drops-104183},
  doi =		{10.4230/LIPIcs.SoCG.2019.14},
  annote =	{Keywords: graph drawing, crossing number, crossing-critical, zip product}
}

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Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12

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