Stars and Bonds in Crossing-Critical Graphs

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Authors

HLINĚNÝ Petr SALAZAR Gelasio

Year of publication 2008
Type Article in Periodical
Magazine / Source Electronic Notes in Discrete Mathematics
MU Faculty or unit

Faculty of Informatics

Citation
Web
Field General mathematics
Keywords crossing number; crossing-critical graph
Description The structure of all known infinite families of crossing--critical graphs has led to the conjecture that crossing--critical graphs have bounded bandwidth. If true, this would imply that crossing--critical graphs have bounded degree, that is, that they cannot contain subdivisions of $K_{1,n}$ for arbitrarily large $n$. In this paper we prove two results that revolve around this conjecture. On the positive side, we show that crossing--critical graphs cannot contain subdivisions of $K_{2,n}$ for arbitrarily large $n$. On the negative side, we show that there are graphs with arbitrarily large maximum degree that are $2$-crossing--critical in the projective plane.
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