The Chromatic Number of the Convex Segment Disjointness Graph

  • Ruy Fabila-Monroy
  • David R. Wood
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let P be a set of n points in general and convex position in the plane. Let D n be the graph whose vertex set is the set of all line segments with endpoints in P, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [CGTA, 2005]. The previous best bounds are \(\frac{3n}{4}\leq \chi(D_n) <n-\sqrt{\frac{n}{2}}\) (ignoring lower order terms). In this paper we improve the lower bound to \(\chi(D_n)\geq n-\sqrt{2n}\), achieving near-tight bounds on χ(D n ).


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  1. 1.
    Araujo, G., Dumitrescu, A., Hurtado, F., Noy, M., Urrutia, J.: On the chromatic number of some geometric type Kneser graphs. Comput. Geom. Theory Appl. 32(1), 59–69 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cairns, G., Nikolayevsky, Y.: Generalized thrackle drawings of non-bipartite graphs. Discrete Comput. Geom. 41(1), 119–134 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cairns, G., Nikolayevsky, Y.: Outerplanar thrackles. Graphs and Combinatorics 28(1), 85–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dujmović, V., Wood, D.R.: Thickness and antithickness (2010) (in preparation)Google Scholar
  6. 6.
    Fenchel, W., Sutherland, J.: Lösung der aufgabe 167. Jahresbericht der Deutschen Mathematiker-Vereinigung 45, 33–35 (1935)Google Scholar
  7. 7.
    Hopf, H., Pammwitz, E.: Aufgabe no. 167. Jahresbericht der Deutschen Mathematiker-Vereinigung 43 (1934)Google Scholar
  8. 8.
    Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18(4), 369–376 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Woodall, D.R.: Thrackles and deadlock. In: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 335–347. Academic Press, London (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ruy Fabila-Monroy
    • 1
  • David R. Wood
    • 2
  1. 1.Departamento de MatemáticasCentro de Investigación y Estudios Avanzados del Instituto Politécnico NacionalMéxico, D.F.México
  2. 2.Department of Mathematics and StatisticsThe Univesity of MelbourneMelbourneAustralia

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