Abstract
We give drawings of the complete graph on orientable and nonorientable surfaces of genus g and improve the best known upper bounds on the crossing number of a complete graph on these surfaces by a factor O(log g). Morover, we give a polynomial time algorithm that produces drawings of arbitrary graphs using the drawings of complete graphs. Using our algorithm we establish an upper bound of O(m2log2g/g) on the crossing number of any graph with n vertices and m edges on an orientable or non-orientable surface of genus g. This upper bound is within a factor of O(log2 g) from the optimal for many classes of graphs.
Research of the third and the fourth author was partially supported by Grant No. 88 of Slovak Academy of Sciences and by EC Cooperative action IC1000 “Algorithms for Future Technologies” (Project ALTEC)
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© 1994 Springer-Verlag Berlin Heidelberg
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Shahrokhi, F., Székely, L.A., Sýkora, O., Vrt'o, I. (1994). Improved bounds for the crossing numbers on surfaces of genus g . In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1993. Lecture Notes in Computer Science, vol 790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57899-4_68
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DOI: https://doi.org/10.1007/3-540-57899-4_68
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