On Degree Properties of Crossing-Critical Families of Graphs
Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs which contain vertices of any prescribed odd degree, for sufficiently large k. From this we derive that, for any set of integers D such that \(\min (D)\ge 3\) and \(3,4\in D\), and for all sufficiently large k there exists a k-crossing-critical family such that the numbers in D are precisely the vertex degrees which occur arbitrarily often in any large enough graph in this family. We also investigate what are the possible average degrees of such crossing-critical families.
KeywordsCrossing number Tile drawing Degree-universality Average degree Crossing-critical graph
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