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

The fractional chromatic number of the plane

  • Daniel W. CranstonEmail author
  • Landon Rabern
Original Paper


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)



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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

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