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@article{1646039,
author = {Bokal, Drago and Bracic, Mojca and Derňár, Marek and Hliněný, Petr},
article_location = {internet},
article_number = {1},
doi = {http://dx.doi.org/10.37236/7753},
keywords = {graph theory; crossing number; crossing-critical},
language = {eng},
issn = {1077-8926},
journal = {Electronic Journal of Combinatorics},
title = {On Degree Properties of Crossing-Critical Families of Graphs},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p53/7812},
volume = {26},
year = {2019}
}
TY - JOUR
ID - 1646039
AU - Bokal, Drago - Bracic, Mojca - Derňár, Marek - Hliněný, Petr
PY - 2019
TI - On Degree Properties of Crossing-Critical Families of Graphs
JF - Electronic Journal of Combinatorics
VL - 26
IS - 1
SP - 1-28
EP - 1-28
PB - -
SN - 10778926
KW - graph theory
KW - crossing number
KW - crossing-critical
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p53/7812
L2 - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p53/7812
N2 - Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large k. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers D such that min(D) >= 3 and 3, 4 is an element of D, and for any sufficiently large k, there exists a k-crossing-critical family such that the numbers in D are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both D and some average degree in the interval (3, 6) are prescribed, k-crossing-critical families exist for any sufficiently large k.
ER -
BOKAL, Drago, Mojca BRACIC, Marek DERŇÁR and Petr HLINĚNÝ. On Degree Properties of Crossing-Critical Families of Graphs. \textit{Electronic Journal of Combinatorics}. internet: -, 2019, vol.~26, No~1, p.~1-28. ISSN~1077-8926. Available from: https://dx.doi.org/10.37236/7753.
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