Crossing number of toroidal graphs

Authors:
János Pach;Géza Tóth
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY;Rényi Institute, Hungarian Academy of Sciences, Budapest, Hungary
Venue:
GD'05 Proceedings of the 13th international conference on Graph Drawing
Year:
2005

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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.