Abstract
We prove that the crossing number of the cartesian product of 2 cycles, C m× Cn, m≤n, is of order Ω(mn), improving the best known lower bound. In particular we show that the crossing number of C m×Cn is at least mn/90, and for n=m, m+1 we reduce the constant 90 to 6. This partially answers a 20-years old question of Harary, Kainen and Schwenk [3] who gave the lower bound m and the upper bound (m−2)n and conjectured that the upper bound is the actual value of the crossing number for C m×Cn. Moreover, we extend this result to k≥3 cycles and paths, and obtain such lower and upper bounds on the crossing numbers of the corresponding meshes, which differ by a small constant only.
Research of the 2nd and the 4th author was partially supported by grant No. 2/1138/94 of Slovak Grant Agency and Alexander von Humboldt Foundation.
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Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt'o, I. (1996). Crossing numbers of meshes. In: Brandenburg, F.J. (eds) Graph Drawing. GD 1995. Lecture Notes in Computer Science, vol 1027. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021830
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DOI: https://doi.org/10.1007/BFb0021830
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