Abstract
In his PhD Thesis E.R. Scheinerman conjectured that planar graphs are intersection graphs of segments in the plane. This conjecture was proved with two different approaches. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.
This research is partially supported by the ANR GATO, under contract ANR-16-CE40-0009.
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References
de Castro, N., Cobos, F., Dana, J.C., Márquez, A., Noy, M.: Triangle-free planar graphs as segment intersection graphs. J. Graph Algorithms Appl. 6(1), 7–26 (2002)
Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane: extended abstract. In: Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing, pp. 631–638 (2009)
Czyzowicz, J., Kranakis, E., Urrutia, J.: A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments. Inform. Process. Lett. 66(3), 125–126 (1998)
Felsner, S.: Triangle contact representations. In: Midsummer Combinatorial Workshop (2009)
de Fraysseix, H., Ossona de Mendez, P.: Representations by contact and intersection of segments. Algorithmica 47, 453–463 (2007)
de Fraysseix, H., de Mendez, P.O., Pach, J.: Representation of planar graphs by segments. Colloq. Math. Soc. János Bolyai 63, 109–117 (1994). Intuitive Geometry (Szeged, 1991)
Gonçalves, D., Isenmann, L., Pennarun, C.: Planar Graphs as L-intersection or L-contact graphs. In: Proceedings of SODA, pp. 172–184 (2018)
Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)
Kratochvíl, J.: String graphs. II. Recognizing string graphs is NP-hard. J. Combin. Theory. Ser. B 52, 67–78 (1991)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory. Ser. B 62, 180–181 (1994)
Máčajová, E., Raspaud, A., Škoviera, M.: The chromatic number of a signed graph. Electr. J. Comb. 23(1), P1, 1–14 (2016)
Pach, J., Solymosi, J.: Crossing patterns of segments. J. Combin. Theory. Ser. A 96, 316–325 (2001)
Scheinerman, E.R.: Intersection classes and multiple intersection parameters of graphs. Ph.D., Thesis, Princeton University (1984)
Scheinerman, E.R.: Private communication to D. West (1993)
West, D.: Open problems. SIAM J. Discrete Math. Newslett. 2(1), 10–12 (1991)
Acknowledgements
The author is thankful to Marc de Visme for fruitful discussions on this topic, and to Pascal Ochem for bringing [11] to his attention.
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Gonçalves, D. (2019). 3-Colorable Planar Graphs Have an Intersection Segment Representation Using 3 Slopes. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_27
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