Advertisement

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

Conference paper
  • 563 Downloads
Part of the CRM Series book series (PSNS, volume 16)

Abstract

We show that every subcubic triangle-free graph has fractional chromatic number at most 14/5, thus confirming a conjecture of Heckman and Thomas [A new proof of the independence ratio of triangle-free cubic graphs. Discrete Math. 233 (2001), 233–237].

Keywords

Planar Graph Maximum Degree Discrete Math Free Graph Fractional Coloring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Albertson, A lower bound for the independence number of a planar graph, J. Combin. Theory Ser. B 20 (1976), 84–93.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    M. Albertson, B. Bollobás and S. Tucker, The independence ratio and maximum degree of a graph, In: “Proceedings of the Seventh Southeastern Conference on Combinatorics”, Graph Theory, and Computing (1976), Utilitas Math., Congressus Numerantium, No. XVII, 43–50.Google Scholar
  3. [3]
    K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977), 429–490.zbMATHMathSciNetGoogle Scholar
  4. [4]
    K. Appel, W. Haken and J. Koch, Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977), 491–567.zbMATHMathSciNetGoogle Scholar
  5. [5]
    Z. Dvořák, J.-S. Sereni and J. Volec, Subcubic triangle-free graphs have fractional chromatic number at most 14/5, submitted for publication, arXiv:1301.5296.Google Scholar
  6. [6]
    S. Fajtlowicz, On the size of independent sets in graphs, Congr. Numer. 21 (1978), 269–274.MathSciNetGoogle Scholar
  7. [7]
    D. Ferguson, T. Kaiser and D. Král’, The fractional chromatic number of triangle-free subcubic graphs, to appear in European Journal of Combinatorics.Google Scholar
  8. [8]
    H. Grötzsch, Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 8 (1958/1959), 109–120.MathSciNetGoogle Scholar
  9. [9]
    H. Hatami and X. Zhu, The fractional chromatic number of graphs of maximum degree at most three, SIAM J. Discrete Math. 23 (2009), 1162–1175.CrossRefMathSciNetGoogle Scholar
  10. [10]
    C. C. Heckman and R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001), 233–237.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    C. C. Heckman and R. Thomas, Independent sets in triangle-free cubic planar graphs, J. Combin. Theory Ser. B 96 (2006), 253–275.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    A. J. W. Hilton, R. Rado and S. H. Scott, A (< 5)-colour theorem for planar graphs, Bull. London Math. Soc. 5 (1973), 302–306.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    K. F. Jones, Independence in graphs with maximum degree four, J. Combin. Theory Ser. B 37 (1984), 254–269.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    K. F. Jones, Size and independence in triangle-free graphs with maximum degree three, J. Graph Theory 14 (1990), 525–535.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    C.-H. Liu, An upper bound on the fractional chromatic number of triangle-free subcubic graphs, submitted for publication.Google Scholar
  16. [16]
    L. Lu and X. Peng, The fractional chromatic number of triangle-free graphs with Δ ≤ 3, Discrete Math. 312 (2012), 3502–3516.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    N. Robertson, D. P. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    W. Staton, Some Ramsey-type numbers and the independence ratio, Trans. Amer. Math. Soc. 256 (1979), 353–370.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    R. Steinberg and C. Tovey, Planar Ramsey numbers, J. Combin. Theory Ser. B 59 (1993), 288–296.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  1. 1.Computer Science Institute of Charles UniversityPragueCzech Republic
  2. 2.Centre National de la Recherche Scientifique (LORIA)NancyFrance
  3. 3.Mathematics Institute and DIMAPUniversity of WarwickCoventryUK

Personalised recommendations