Abstract
We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graphT=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple, if any two edges meet in at most one common point.
Let h=h(n) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω(n 3/2)≤ h(n) ≤ O(n 2/log1/4 n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn 2.
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References
Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Annals of Discrete Mathematics 12, 9–12 (1982)
Brass, P., Moser, W., Pach, J.: Research problems in discrete geometry. Springer, Heidelberg (2005)
Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23, 191–206 (2000)
Černý, J.: Geometric graphs with no three disjoint edges. Discrete Comput. Geom. (to appear)
Harborth, H.: Crossings on edges in drawings of complete multipartite graphs. Colloquia Math. Soc. János Bolyai 18, 539–551 (1978)
Harborth, H., Mengersen, M.: Edges without crossings in drawings of complete graphs. J. Comb. Theory, Ser. B 17, 299–311 (1974)
Harborth, H., Mengersen, M.: Drawings of the complete graph with maximum number of crossings. Congr. Numerantium 88, 225–228 (1992)
Harborth, H., Thürmann, C.: Minimum number of edges with at most s crossings in drawings of the complete graph. Congr. Numerantium 102, 83–90 (1994)
Leighton, F.T.: New lower bound techniques for VLSI. Math. Systems Theory 17, 47–70 (1984)
Pach, J., Pinchasi, R., Tardos, G., Tóth, G.: Geometric graphs with no self-intersecting path of length three. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 295–311. Springer, Heidelberg (2002); also European J. Combin. 25(6), 793–811 (2004)
Pach, J., Radoičić, R., Tóth, G.: A generalization of quasi-planarity. In: Towards a theory of geometric graphs. Contemp. Math., vol. 342, pp. 177–183. Amer. Math. Soc, Providence (2004)
Pach, J., Radoičić, R., Tóth, G.: Relaxing planarity for topological graphs. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 221–232. Springer, Heidelberg (2003)
Pach, J., Solymosi, J., Tóth, G.: Unavoidable configurations in topological complete graphs. Discrete Comput. Geom. 30, 311–320 (2003)
Pach, J., Tóth, G.: Disjoint edges in topological graphs (to appear)
Pinchasi, R., Radoičić, R.: Topological graphs with no self-intersecting cycle of length 4. In: Towards a theory of geometric graphs. Contemp. Math., vol. 342, pp. 233–243. Amer. Math. Soc., Providence (2004)
Ringel, G.: Extremal problems in the theory of graphs. In: Theory Graphs Appl., Proc. Symp. Smolenice 1963, pp. 85–90 (1964)
Ringeisen, R.D., Stueckle, S.K., Piazza, B.L.: Subgraphs and bounds on maximum crossings. Bull. Inst. Comb. Appl. 2, 33–46 (1991)
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Kynčl, J., Valtr, P. (2006). On Edges Crossing Few Other Edges in Simple Topological Complete Graphs. In: Healy, P., Nikolov, N.S. (eds) Graph Drawing. GD 2005. Lecture Notes in Computer Science, vol 3843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11618058_25
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DOI: https://doi.org/10.1007/11618058_25
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