Graphs and Combinatorics

, Volume 30, Issue 1, pp 71–81 | Cite as

On the Chromatic Number of Subsets of the Euclidean Plane

  • M. AxenovichEmail author
  • J. Choi
  • M. Lastrina
  • T. McKay
  • J. Smith
  • B. Stanton
Original Paper


The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is at least 4 and at most 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines.


Chromatic number of the plane Unit distance graph Coloring the plane 

Mathematics Subject Classification (1991)

Primary 05C12 05C15 05C62 51K99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauslaugh B.L.: Tearing a strip off the plane. J. Graph Theory 29(1), 17–33 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Boreico I.: My favorite problem: linear independence of radicals. Harv. Coll. Math. Rev. 2, 87–92 (2008)Google Scholar
  3. 3.
    Braß P., Pach J.: The maximum number of times the same distance can occur among the vertices of a convex n-gon is O(nlog n). J. Combin. Theory Ser. A 94(1), 178–179 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chilakamarri K.B.: The unit-distance graph problem: a brief survey and some new results. Bull. Inst. Combin. Appl. 8, 39–60 (1993)zbMATHMathSciNetGoogle Scholar
  5. 5.
    de Bruijn, N.G., Erdős, P.: A colour problem for infinite graphs and a problem in the theory of relations. em Nederl. Akad. Wetensch. Proc. Ser. A. vol. 54. Indagationes Math. 13, 369–373 (1951)Google Scholar
  6. 6.
    Edelsbrunner P., Hajnal H.: A lower bound on the number of unit distances between the vertices of a convex polygon. J. Combin. Theory Ser. A 56(2), 312–316 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Erdős, P., Fishburn, P.C.: Multiplicities of interpoint distances in finite planar sets. Discrete Appl. Math. 60(1–3), 141–147 (1995) (ARIDAM VI and VII (New Brunswick, NJ, 1991/1992))Google Scholar
  8. 8.
    Erdős P., Moser, L.: Problem 11. Can. Math. Bull. 2, 43 (1959)Google Scholar
  9. 9.
    Fischer K.G.: Additive K colorable extensions of the rational plane. Discrete Math. 82(2), 181–195 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Füredi Z.F.: The maximum number of unit distances in a convex n-gon. J. Combin. Theory Ser. 55(2), 316–320 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Johnson, Jr. P.D.: Two-colorings of real quadratic extensions of Q 2 that forbid many distances. Congr. Numer. 60, 51–58 (1987). Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1987).Google Scholar
  12. 12.
    Klazar, M.: A question on linear independence of square roots (Unpublished)Google Scholar
  13. 13.
    Kruskal C.P.: The chromatic number of the plane: the bounded case. J. Comput. Syst. Sci. 74(4), 598–627 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Moorehouse, G.E.: On the chromatic numbers of planes (Preprint)Google Scholar
  15. 15.
    Pritikin, D.: All unit-distance graphs of order 6197 are 6-colorable. J. Combin. Theory Ser. B 73(2), 159–163 (1998)Google Scholar
  16. 16.
    Soifer, A.: The mathematical coloring book. Springer, New York (2009) (Mathematics of coloring and the colorful life of its creators, With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau)Google Scholar
  17. 17.
    Woodall D.R.: Distances realizes by sets covering the plane. J. Combin. Theory Ser. A 14, 187–200 (1973)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2012

Authors and Affiliations

  • M. Axenovich
    • 1
    Email author
  • J. Choi
    • 1
  • M. Lastrina
    • 1
  • T. McKay
    • 1
  • J. Smith
    • 1
  • B. Stanton
    • 1
  1. 1.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations