Scheinerman's conjecture

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In mathematics, Scheinerman's conjecture states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich et al. 1976).

For instance, the graph G shown below to the left may be represented as the intersection graph of the set of segments shown below to the right. Here, vertices of G are represented by straight line segments and edges of G are represented by intersection points.

 

Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and D. West (1991) conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directions and no two segments belong to the same line, then the graph can be colored using k colors, one color for each direction. Therefore, if every planar graph can be represented in this way with only four directions, then the four color theorem follows.

Hartman et al. (1991) and de Fraisseix et al. (1991) proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also Czyzowicz et al (1998). De Castro et al. (2002) proved that every triangle-free planar graph can be represented as an intersection graph of line segments having only three directions; this result implies Grötzsch's theorem (Grötzsch 1959) that triangle-free planar graphs can be colored with three colors. De Fraysseix and Ossona de Mendez (2005) proved that if a planar graph G can be 4-colored in such a way that no separating cycle uses all four colors, then G has a representation as an intersection graph of segments.

Chalopin et al. (2007) proved that planar graphs are in 1-STRING, the class of intersection graphs of simple curves in the plane that intersect each other in at most one crossing point per pair. This class is intermediate between the intersection graphs of segments appearing in Scheinerman's conjecture and the intersection graphs of unrestricted simple curves from the result of Ehrlich et al.

[edit] References

  • Chalopin, J.; Gonçalves, D.; Ochem, P. (2007). "Planar graphs are in 1-STRING". Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms: 609–617, ACM and SIAM. 
  • Czyzowicz, J.; Kranakis, E.; Urrutia, J. (1998). "A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments". Information Processing Letters 66 (3): 125–126. doi:10.1016/S0020-0190(98)00046-5. 
  • de Fraysseix, H.; Ossona de Mendez, P.; Pach, J. (1991). "Representation of planar graphs by segments". Intuitive Geometry 63: 109–117. MR1383616. 
  • Grötzsch, Herbert (1959). "Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel". Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120. MR0116320. 
  • Scheinerman, E. R. (1984). "Intersection classes and multiple intersection parameters of graphs". Ph.D. thesis. . Princeton University
  • West, D. (1991). "Open problems". SIAM J. Discrete Math. Newslett. 2 (1): 10–12.