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Expected Crossing Numbers

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

Abstract

The expected value for the weighted crossing number of a randomly weighted graph is studied. We focus on the case where G = K n and the edge-weights are independent random variables that are uniformly distributed on [0,1]. The first non-trivial case is K 5. We compute this via an unexpectedly involved calculation, and consider bounds for larger values of n. A variation of the Crossing Lemma for expectations is proved.

Key words

Graph Crossing number Weighted crossing number Crossing lemma 

Subject Classifications

05C10 60C05 

Notes

Acknowledgements

An extended abstract containing preliminary versions of these results appeared in the proceedings of EUROCOMB 2011 [9].

B. Mohar was supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). He is on leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. T. Stephen was supported in part by an NSERC Discovery Grant. The authors are grateful to Luis Goddyn for some helpful discussions on the subject, and to the anonymous referees for helpful comments.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.CRC in Graph Theory, Tier 1, Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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