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Two maps on one surface

โœ Scribed by Dan Archdeacon; C. Paul Bonnington

Publisher
John Wiley and Sons
Year
2001
Tongue
English
Weight
193 KB
Volume
36
Category
Article
ISSN
0364-9024

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โœฆ Synopsis

Two embeddings of a graph in a surface S are said to be equivalent'' if they are identical under an homeomorphism of S that is orientation-preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be jointly embedded'' if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to ยฎnd equivalent embeddings of the two graphs that minimize the number of these edge-crossings; this minimum we call the joint crossing number'' of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re-embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its mirror image,'' then the joint crossing number can decrease, but not by more than 6.066%.

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