Abstract
The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n× n grid with no three points collinear. In 1951, Erdös proved that the answer is Θ(n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n× n× n grid with no three collinear is Θ(n 2). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in ℤ3, such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph K n is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of K n is Θ(n 3/2). This compares favourably to Θ(n 3) when edges are not allowed to cross. Generalising the construction for K n , we prove that every k-colourable graph on n vertices has a 3D drawing with \(\mathcal{O}(n\sqrt{k})\) volume. For the k-partite Turán graph, we prove a lower bound of Ω((kn)3/4).
Research supported by NSERC and COMBSTRU.
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References
Adena, M.A., Holton, D.A., Kelly, P.A.: Some thoughts on the no-three-in-line problem. In: Proc. 2nd Australian Conf. on Combinatorial Mathematics. Lecture Notes in Math., vol. 403, pp. 6–17. Springer, Heidelberg (1974)
Anderson, D.B.: Update on the no-three-in-line problem. J. Combin. Theory Ser. A 27(3), 365–366 (1979)
Baker, R.C., Harman, G., Pintz, J.: The difference between consecutive primes. II. Proc. London Math. Soc. 83(3), 532–562 (2001)
Bose, P., Czyzowicz, J., Morin, P., Wood, D.R.: The maximum number of edges in a three-dimensional grid-drawing. J. Graph Algorithms Appl. 8(1), 21–26 (2004)
Calamoneri, T., Sterbini, A.: 3D straight-line grid drawing of 4-colorable graphs. Inform. Process. Lett. 63(2), 97–102 (1997)
Cohen, R.F., Eades, P., Lin, T., Ruskey, F.: Threedimensional graph drawing. Algorithmica 17(2), 199–208 (1996)
Craggs, D., Hughes-Jones, R.: On the no-three-in-line problem. J. Combinatorial Theory Ser. A 20(3), 363–364 (1976)
Di Giacomo, E.: Drawing series-parallel graphs on restricted integer 3D grids. In: Liotta [22], pp. 238–246
Di Giacomo, E., Meijer, H.: Track drawings of graphs with constant queue number. In: Liotta [22], pp. 214–225
Dudeney, H.E.: Amusements in Mathematics. Nelson, Edinburgh (1917)
Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. (to appear)
Dujmović, V., Wood, D.R.: Three-dimensional grid drawings with subquadratic volume. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 55–66. Amer. Math. Soc. (2004)
Erdös, P.: A theorem of Sylvester and Schur. J. London Math. Soc. 9, 282–288 (1934)
Erdös, P.: Appendix, in Klaus F. Roth. On a problem of Heilbronn. J. London Math. Soc. 26, 198–204 (1951)
Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7(4), 363–398 (2003)
Flammenkamp, A.: Progress in the no-three-in-line problem. II. J. Combin. Theory Ser. A 81(1), 108–113 (1998)
Guy, R.K., Kelly, P.A.: The no-three-in-line problem. Canad. Math. Bull. 11, 527–531 (1968)
Hall, R.R., Jackson, T.H., Sudbery, A., Wild, K.: Some advances in the no-three-in-line problem. J. Combinatorial Theory Ser. A 18, 336–341 (1975)
Harborth, H., Oertel, P., Prellberg, T.: No-three-in-line for seventeen and nineteen. Discrete Math. 73(1-2), 89–90 (1989)
Hasunuma, T.: Laying out iterated line digraphs using queues. In: Liotta [22], pp. 202–213
Kløve, T.: On the no-three-in-line problem. III. J. Combin. Theory Ser. A 26(1), 82–83 (1979)
Liotta, G. (ed.): GD 2003. LNCS, vol. 2912. Springer, Heidelberg (2004)
Pach, J., Thiele, T., Tóth, G.: Three-dimensional grid drawings of graphs. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in discrete and computational geometry. Contemporary Mathematics, vol. 223, pp. 251–255. Amer. Math. Soc. (1999)
Wood, D.R.: Drawing a graph in a hypercube. Manuscript (2004)
Wood, D.R.: Grid drawings of k-colourable graphs. Comput. Geom. (to appear)
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Pór, A., Wood, D.R. (2005). No-Three-in-Line-in-3D. In: Pach, J. (eds) Graph Drawing. GD 2004. Lecture Notes in Computer Science, vol 3383. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31843-9_40
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DOI: https://doi.org/10.1007/978-3-540-31843-9_40
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