, Volume 37, Issue 5, pp 863–886 | Cite as

Density of 5/2-critical graphs

  • Zdeněk DvořákEmail author
  • Luke Postle
Original Paper


A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C 5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n ≥ 4 vertices has at least
$$\frac{{5n - 2}}{4}$$
edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.

Mathematics Subject Classification (2000)



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  1. [1]
    O. Borodin, S. G. Hartke, A. O. Ivanova, A. V. Kostochka and D. B. West: Circular (5,2)-coloring of sparse graphs, Sib. Elektron. Mat. Izv. 5 (2008), 417–426.MathSciNetzbMATHGoogle Scholar
  2. [2]
    O. Borodin, S.-J. Kim, A. Kostochka and D. West: Homomorphisms from sparse graphs with large girth, Journal of Combinatorial Theory, Series B 90 (2004), 147–159.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. DeVos: Open problem garden: Mapping planar graphs to odd cycles Scholar
  4. [4]
    H. Grötzsch: Ein Dreifarbensatz für Dreikreisfreie Netze auf der Kugel, Math.Natur. Reihe 8 (1959), 109–120.MathSciNetGoogle Scholar
  5. [5]
    F. Jaeger: On circular ows in graphs, in: Finite and Infinite Sets (Eger, 1981), Colloq. Math. Soc. J. Bolyai, vol. 37, 1984, 391–402.CrossRefGoogle Scholar
  6. [6]
    W. Klostermeyer and C. Q. Zhang: (2+ε)-coloring of planar graphs with large odd-girth, J. Graph Theory 33 (2000), 109–119.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Kostochka and M. Yancey: Ore’s conjecture for k=4 and Grötzsch Theorem, Manuscript, 2012.zbMATHGoogle Scholar
  8. [8]
    O. Ore: The Four Color Problem, Academic Press, New York, 1967.zbMATHGoogle Scholar
  9. [9]
    A. Vince: Star chromatic number, Journal of Graph Theory 12 (1988), 551–559.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    X. Zhu: Circular chromatic number: a survey, Discrete Mathematics 229 (2001), 371–410.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    X. Zhu: Circular chromatic number of planar graphs of large odd girth, Electronic J. of Combinatorics 8 (2001), R25.MathSciNetzbMATHGoogle Scholar
  12. [12]
    X. Zhu: Recent developments in circular colouring of graphs, in: Topics in discrete mathematics, Springer, 2006, 497–550.CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Charles UniversityPragueCzech Republic
  2. 2.University of WaterlooWaterlooCanada

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