Advertisement

Combinatorica

, Volume 37, Issue 5, pp 837–861 | Cite as

The fractional chromatic number of the plane

  • Daniel W. CranstonEmail author
  • Landon Rabern
Original Paper

Abstract

The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted χ(ℝ2). The problem was introduced in 1950, and shortly thereafter it was proved that 4≤χ(ℝ2)χ≤7. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate χ f (ℝ2), the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were 3.5556≤χ f (ℝ2)≤4.3599. Here we improve the lower bound to 76/21≈3.6190.

Mathematics Subject Classification (2000)

05C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Croft: Incidence Incidents, Eureka 30 (1967), 22–26.Google Scholar
  2. [2]
    N. G. de Bruijn and P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 369–373.MathSciNetzbMATHGoogle Scholar
  3. [3]
    P. Erdős: Unsolved problems, Congressus Numerantium XV–Proceedings of the 5th British Combinatorial Conference. 1975, 681, 1976.Google Scholar
  4. [4]
    P. Erdős: Problems and results in combinatorial geometry, In Discrete geometry and convexity (New York, 1982), volume 440 of Ann. New York Acad. Sci., pages 1–11. New York Acad. Sci., New York, 1985.Google Scholar
  5. [5]
    F. M. de Oliveira Filho and F. Vallentin: Fourier analysis, linear programming, and densities of distance avoiding sets in Rn, Journal of the European Mathematical Society, 12 (2010), 1417–1428.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Falconer: The realization of distances in measurable subsets covering Rn, Journal of Combinatorial Theory, Series A, 31 (1981), 184–189.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    K. Fischer: Additive k-colorable extensions of the rational plane, Discrete Mathematics 82 (1990), 181–195.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D. Fisher and D. Ullman: The fractional chromatic number of the plane, Geombinatorics 2 (1992), 8–12.MathSciNetzbMATHGoogle Scholar
  9. [9]
    M. Gardner: Mathematical games, Scientific American, 206:172–180, 10 1960.Google Scholar
  10. [10]
    J. Grytczuk, K. Junosza-Szaniawski, J. Sokół and K. Wesek: Fractional and j-fold colouring of the plane, Arxiv preprint: http://arxiv.org/abs/1506.01887.Google Scholar
  11. [11]
    H. Hadwiger: Ueberdeckung des Euklidischen Raumes durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.MathSciNetzbMATHGoogle Scholar
  12. [12]
    R. Hochberg and P. O’Donnell: A large independent set in the unit distance graph, Geombinatorics 2 (1993), 83–84.MathSciNetzbMATHGoogle Scholar
  13. [13]
    G. M. Levin: Selected Topics in Fractional Graph Theory, 1997, Thesis (Ph.D.)–The Johns Hopkins University.Google Scholar
  14. [14]
    S. L. Mahan: The fractional chromatic number of the plane, Master’s thesis, University of Colorado Denver, 1995.Google Scholar
  15. [15]
    B. Mohar: http://www.fmf.uni-lj.si/~mohar/Problems/P8UnitDistanceGraph.html.Google Scholar
  16. [16]
    L. Moser and W. Moser: Solution to problem 10, Can. Math. Bull. 4 (1961), 187–189.zbMATHGoogle Scholar
  17. [17]
    P. M. O’Donnell: High-girth unit-distance graphs, ProQuest LLC, Ann Arbor, MI, 1999, Thesis (Ph.D.)–Rutgers The State University of New Jersey-New Brunswick.Google Scholar
  18. [18]
    D. Pritikin: All unit-distance graphs of order 6197 are 6-colorable, Journal of Combinatorial Theory, Series B 73 (1998), 159–163.MathSciNetzbMATHGoogle Scholar
  19. [19]
    E. R. Scheinerman and D. H. Ullman: Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1997, A rational approach to the theory of graphs, with a foreword by Claude Berge, A Wiley-Interscience Publication.zbMATHGoogle Scholar
  20. [20]
    A. Soifer: 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.zbMATHGoogle Scholar
  21. [21]
    R. Solovay: A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics (1970), 1–56.Google Scholar
  22. [22]
    L. Székely: Measurable chromatic number of geometric graphs and sets without some distances in euclidean space, Combinatorica 4 (1984), 213–218.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    D. Woodall: Distances realized by sets covering the plane, Journal of Combinatorial Theory, Series A 14 (1973), 187–200.zbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  2. 2.Department of MathematicsFranklin & Marshall CollegeLancasterUSA

Personalised recommendations