Skip to main content
SpringerLink
Log in

Relaxed chromatic numbers of graphs

  • Original Papers
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Given any family of graphsP, theP chromatic number χ p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akiyama, J., Era, H., Gervacio, S.V., Watanabe, M.: Path chromatic numbers of graphs. J. Graph Theory13, 569–575 (1989).

    Google Scholar 

  2. Broere, I., Frick, M.: On the order of color critical graphs. In: Proceedings of the SE Conference on Graph Th., Comb., and Comp., Congr. Number.47, 125–130 (1985)

  3. Broere, I., Frick, M.: On the order of uniquely colorable graphs. Verslagreeks van die Departement Wiskunde 8/86 (1986)

    Google Scholar 

  4. Broere, I., Frick, M.: A characterization of the sequence of generalized chromatic numbers of a graph. Res. Rept. 64/88(9), Univ. of S. Africa (1988)

  5. Brooks, R.L.: On colouring the nodes of a network. Proc. Cambridge Philos. Soc.37, 194–197 (1941)

    Google Scholar 

  6. Brown, J.I.: OnP colorings of graphs. Ph.D. Thesis, University of Waterloo, 1988

  7. Burr, S.A., Jacobson, M.S.: On inequalities involving vertex-partition parameters of graphs. Congr. Numerantium70, 159–170 (1990)

    Google Scholar 

  8. Hakimi, S.L., et al. on vertex arboricity

  9. Jacob, H., Meyniel, H.: Extensions of Tur'an's and Brooks' Theorems and new notions of stability and coloring in digraphs. Ann. Discrete Math.17, 365–370 (1983)

    Google Scholar 

  10. Johns, G., Saba, F.: On the path-chromatic number of a graph. In: Graph Theory and Its Applications: East and West (Proc. 1st China-U.S.A. Intl. Conf. Gr. Th., Jinan, 1986) Annals N.Y. Acad. Sci576, 275–280 (1989).

  11. Jones, R.P.: Hereditary properties and P-chromatic numbers, In: Combinatorics (Proc. Brit. Comb. Conf. 1973) McDonough, T.P., Marron, V.C. (eds.), Cambridge Univ. Press, 83–88 (1974).

  12. Lovász, L.: On decomposition of graphs. Studia Sci. Math. Hungar.1, 237–238 (1966)

    Google Scholar 

  13. Matula, D.W.: An extension of Brooks' theorem. Research report, Center for Numerical Analysis, Univ. of Texas, Austin (1973)

    Google Scholar 

  14. Schäuble, M.: Benerkungen zur Konstruktion dreikreisfreierk-chromatischer Graphen. Wiss. Z. Tech Hochsch Ilmenau15 59–63 (1969)

    Google Scholar 

  15. Szekeres G., Wilf, H.S.: An inequality for the chromatic number of a graph. J. Comb. Theory4, 1–3 (1968)

    Google Scholar 

  16. Toft, B.: Graph colouring theory. In: Graham, R.L., Lovász, L., Schrijver, A. (eds.): The Handbook of Combinatorics, to be published

  17. Zykov, A.A.: On some properties of linear complexes (in Russian). Mat. Sbornik N.S.24, 163–168 (1949). English translation in Amer. Math. Soc. Transl. 79(1952), reissued in Translations Series 1 Vol 7 Algebraic Topology (Amer. Math. Soc., 1962)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Eastern Illinois University, 61920, Charleston, IL, USA

    Margaret L. Weaver

  2. University of Illinois, 61801-2975, Urbana, IL, USA

    Douglas B. West

Authors
  1. Margaret L. Weaver
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Douglas B. West
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by ONR Grant N00014-85K0570 and by a grant from the University of Illinois Research Board.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Weaver, M.L., West, D.B. Relaxed chromatic numbers of graphs. Graphs and Combinatorics 10, 75–93 (1994). https://doi.org/10.1007/BF01202473

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01202473

Keywords

Search

Navigation