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3-Colorable Planar Graphs Have an Intersection Segment Representation Using 3 Slopes

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11789)


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.


  • Planar graphs
  • Segment intersections

This research is partially supported by the ANR GATO, under contract ANR-16-CE40-0009.

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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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Correspondence to Daniel Gonçalves .

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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.

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