Abstract
It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant. This bound, conjectured by Brass, cannot be improved, apart from the value of the constant. We strengthen and generalize this result to the case when the graph has a crossing-free drawing on an orientable surface of higher genus and there is no restriction on the degrees of the vertices.
János Pach has been supported by NSF Grant CCR-00-98246, and by grants from PSC-CUNY, OTKA, NSA, and BSF. Géza Tóth has been supported by OTKA-T-038397 and T-046246.
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© 2006 Springer-Verlag Berlin Heidelberg
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Pach, J., Tóth, G. (2006). Crossing Number of Toroidal Graphs. In: Healy, P., Nikolov, N.S. (eds) Graph Drawing. GD 2005. Lecture Notes in Computer Science, vol 3843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11618058_30
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DOI: https://doi.org/10.1007/11618058_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31425-7
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