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DOI: 10.4230/LIPIcs.SoCG.2019.14
URN: urn:nbn:de:0030-drops-104183
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10418/
Bokal, Drago ; Dvorák, Zdenek ; Hlinený, Petr ; Leaños, Jesús ; Mohar, Bojan ; Wiedera, Tilo
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: | 11.06.2019 |
