In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of in a surface with Euler genus ε is ⎣n/(2ε+2)⎦ n - (ε+1)(1+⎣n/(2ε+2)⎦).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1379, author = {Pak Tung Ho}, title = {A proof of the crossing number of $K\_{3,n}$ in a surface}, journal = {Discussiones Mathematicae Graph Theory}, volume = {27}, year = {2007}, pages = {549-551}, zbl = {1142.05018}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1379} }
Pak Tung Ho. A proof of the crossing number of $K_{3,n}$ in a surface. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 549-551. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1379/
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