Advertisement

A Chromatic Metric on Graphs

Chapter

Abstract

In this chapter, we introduce the concept of relatedness of graphs, based upon the generalized chromatic number. This allows the definition of a graph metric. It is proved that the distance between any two graphs is at most three.

Keywords

Graphs Chromatic number Generalized graph coloring Graph distance Metric 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benadé JG (1990) Some aspects of generalised graph colourings. Ph.D thesis, Department of Mathematics, Rand Afrikaans UniversityGoogle Scholar
  2. 2.
    Benadé G, Broere I (1988) Generalized colourings: existence of uniquely colourable graphs. Verslagreeks van die Departement Wiskunde, RAU, no. 6/88Google Scholar
  3. 3.
    Benadé G, Broere I (1990) A construction of uniquely C 4-free colourable graphs. Quaest Math 13:259–264MATHCrossRefGoogle Scholar
  4. 4.
    Benadé G, Broere I (1992) Chromatic relatedness of graphs. Ars Combinatoria 34:326–330MATHMathSciNetGoogle Scholar
  5. 5.
    Bollobás B, Thomason AG (1977) Uniquely partitionable graphs. J Lond Math Soc 16: 403–410MATHCrossRefGoogle Scholar
  6. 6.
    Broere I, Frick M (1988) A characterization of the sequence of generalized chromatic numbers of a graph. In: Proceedings of the 6th international conference on the theory and applications of graphs, KalamazooGoogle Scholar
  7. 7.
    Broere I, Frick M (1990) On the order of uniquely colourable graphs. Discrete Math 82(3): 225–232MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Brown JI (1987) A theory of generalized graph colouring. Ph.D. thesis, Department of Mathematics, University of TorontoGoogle Scholar
  9. 9.
    Brown JI, Corneil DG (1987) On generalized graph colourings. J Graph Theory 11:87–99MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Burger M (1984) The cochromatic number of a graph. Ph.D. thesis, Department of Mathematics, Rand Afrikaans UniversityGoogle Scholar
  11. 11.
    Chartrand G, Geller DP (1968) On uniquely colorable planar graphs. J Combin Theory 6: 265–271Google Scholar
  12. 12.
    Chartrand G, Lesniak L (1986) Graphs and digraphs, 2nd edn. Wadsworth, BelmontMATHGoogle Scholar
  13. 13.
    Folkman J (1970) Graphs with monochromatic complete subgraphs in every edge colouring. SIAM J Appl Math 18:19–24MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Frick M (1987) Generalised colourings of graphs. Ph.D thesis, Department of Mathematics, Rand Afrikaans UniversityGoogle Scholar
  15. 15.
    Greenwell D, Lovász L (1974) Applications of product colouring. Acta Math Acad Sci H 25:335–340MATHCrossRefGoogle Scholar
  16. 16.
    Harary F (1985) Conditional colorablility of graphs. In: Harary F, Maybee J (eds) Graphs and applications. Proc 1st Col Symp Graph theory. Wiley, New York, pp 127–136Google Scholar
  17. 17.
    Lesniak-Foster L, Straight HJ (1977) The cochromatic number of a graph. Ars Combinatoria 3:39–45MATHMathSciNetGoogle Scholar
  18. 18.
    Mynhardt CM, Broere I (1985) Generalized colorings of graphs. In: Alavi Y et al (eds) Graph theory and its applications to algorithms and computer science. Wiley, New York, pp 583–594Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Computer Science, Statistics and MathematicsNorth-West UniversityPotchefstroomSouth Africa

Personalised recommendations