Abstract
Let G be a graph with n vertices and e≥4n edges, drawn in the plane in such a way that if two or more edges (arcs) share an interior point p, then they properly cross one another at p. It is shown that the number of crossing points, counted without multiplicity, is at least constant times e and that the order of magnitude of this bound cannot be improved. If, in addition, two edges are allowed to cross only at most once, then the number of crossing points must exceed constant times (e/n)4.
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Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. In: Theory and Practice of Combinatorics. North-Holland Math. Stud., vol. 60, pp. 9–12. North-Holland, Amsterdam (1982)
Kleitman, D.: The crossing number of K 5,n . J. Comb. Theory 9, 315–323 (1970)
Leighton, F.T.: Complexity Issues in VLSI: Optimal Layouts for the Shuffle-Exchange Graph and Other Networks. MIT Press, Cambridge (1983)
Mohar, B.: Personal communication
Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)
Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000)
Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16, 111–117 (1996)
Pach, J., Spencer, J., Tóth, G.: New bounds on crossing numbers. Discrete Comput. Geom. 24, 623–644 (2000)
Székely, L.: Crossing numbers and hard Erdős problems in discrete geometry. Comb. Probab. Comput. 6, 353–358 (1997)
Szemerédi, E., Trotter, W.T.: Extremal problems in discrete geometry. Combinatorica 3, 381–392 (1983)
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The research of J. Pach was supported by NSF grant CCF-05-14079 and by grants from NSA, PSC-CUNY, BSF, and OTKA-K-60427.
The research of G. Tóth was supported by OTKA-K-60427.
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Pach, J., Tóth, G. Degenerate Crossing Numbers. Discrete Comput Geom 41, 376–384 (2009). https://doi.org/10.1007/s00454-009-9141-y
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DOI: https://doi.org/10.1007/s00454-009-9141-y