Abstract
This paper studies the question: What is the maximum integer k b,n such that every k b,n-colorable graph has a b-bend n-dimensional orthogonal box drawing?
We give an exact answer for the orthogonal line drawing in all dimensions and for the 3-dimensional rectangle visibility representation. We present an upper and lower bound for the 3-dimensional orthogonal drawing by rectangles and general boxes. Particularly, we improve the best known upper bound for the 3-dimensional orthogonal box drawing from 183 to 42 and the lower bound from 3 to 22.
Supported by the Institute for Theoretical Computer Science, Charles University, Prague, project No. 1M0545 of the Czech Ministry of Education.
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Wood, D.R.: Three-Dimensional Orthogonal Graph Drawing, Ph.D. Thesis, School of Computer Science and Software Engineering, Monash University (2000)
Fekete, S.P., Meijer, H.: Rectangle and box visibility graphs in 3D. Int. J. Comput. Geom. Appl. 97, 1–28 (1997)
Fekete, S.P., Houle, M.E., Whitesides, S.: New results on a visibility representation of graphs in 3D. In: Proc. Graph Drawing, vol. 95, pp. 234–241 (1995)
Bose, P., Dean, A., Hutchinson, J., Shermer, T.: On rectangle visibility graphs. In: Proc. Graph Drawing, vol. 96, pp. 25–44 (1996)
Hutchinson, J.P., Shermer, T., Vince, A.: On representations of some thickness-two graphs. Comput. Geom. 13(3), 161–171 (1999)
Dean, A.M., Hutchinson, J.P.: Rectangle-visibility Layouts of Unions and Products of Trees. J. Graph Algorithms Appl., 1–21 (1998)
Shermer, T.: Block visibility representations III: External visibility and complexity. In: Fiala, K., Sack (eds.) Proc. of 8th Canadian Conf. on Comput. Geom. Int. Informatics Series, vol. 5, pp. 234–239. Carleton University Press, Ottawa (1996)
Fekete, S.P., Houle, M.E., Whitesides, S.:: The wobbly logic engine: proving hardness of non-rigid geometric graph representation problems, Report No. 97.273, Angewandte Mathematik und Informatik Universität zu Köln (1997)
Štola, J.: Chromatic invariants in graph drawing, Master’s Thesis, Department of Applied Mathematics, Charles University, Prague, Czech Republic (2006), http://kam.mff.cuni.cz/~stola/chromaticInvariants.pdf
Chvátal, V.: On finite polarized partition relations. Canad. Math. Bul. 12, 321–326 (1969)
Beineke, L.W., Schwenk, A.J.: On a bipartite form of the Ramsey problem. In: Proc. 5th British Combin. Conf. 1975, Congr. Numer., vol. XV, pp. 17–22 (1975)
Biedl, T., Shermer, T., Whitesides, S., Wismath, S.: Bounds for orthogonal 3D graph drawing. J. Graph Alg. Appl. 3(4), 63–79 (1999)
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Štola, J. (2008). Colorability in Orthogonal Graph Drawing. In: Hong, SH., Nishizeki, T., Quan, W. (eds) Graph Drawing. GD 2007. Lecture Notes in Computer Science, vol 4875. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77537-9_32
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DOI: https://doi.org/10.1007/978-3-540-77537-9_32
Publisher Name: Springer, Berlin, Heidelberg
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