Colorings and orientations of graphs


Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon, S. Friedland, andG. Kalai: Regular subgraphs of almost regular graphs,J. Combinatorial Theory, Ser. B37 (1984), 79–91.

    MathSciNet  Article  Google Scholar 

  2. [2]

    N. Alon, C. McDiarmid, andB. Reed: Star Arboricity, to appear.

  3. [3]

    N. Alon, andM. Tarsi: A nowhere zero point in linear mappings,Combinatorica 9 (1989), 393–395.

    MathSciNet  Article  Google Scholar 

  4. [4]

    J. A. Bondy, R. Boppana, andA. Siegel: Private communication.

  5. [5]

    C. Berge:Graphs and Hypergraphs, Dunod, Paris, 1970.

    Google Scholar 

  6. [6]

    B. Bollobás:Extremal Graph Theory, Academic Press, New York, 1978.

    Google Scholar 

  7. [7]

    A. Chetwynd, andR. Häggkvist: A note on list colorings,J. Graph Theory 13 (1989), 87–95.

    MathSciNet  Article  Google Scholar 

  8. [8]

    P. Erdős: Some old and new problems in various branches of combinatorics,Congressus Numerantium 23 (1979), 19–37.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    P. Erdős, A. Rubin, andH. Taylor: Choosability in graphs,Congressus Numerantium 26 (1979), 125–157.

    MATH  Google Scholar 

  10. [10]

    I. Gessel: Tournaments and Vandermonde's determinant,J. Graph Theory 3 (1979), 305–307.

    MathSciNet  Article  Google Scholar 

  11. [11]

    R. L. Graham, S.-Y. R. Li, andW.-C. W. Li: On the structure oft-designs,SIAM J. Alg. Disc. Meth. 1 (1980), 8–14.

    Article  Google Scholar 

  12. [12]

    S.-Y. R. Li, andW.-C. W. Li: Independence numbers of graphs and generators of ideals,Combinatorical 1 (1981), 55–61.

    MathSciNet  Article  Google Scholar 

  13. [13]

    R. P. Stanley: Acyclic orientations of graphs,Discrete Math. 5 (1973), 171–178.

    MathSciNet  Article  Google Scholar 

  14. [14]

    M. Tarsi: On the decomposition of a graph into stars,Discrete Math. 36 (1981), 299–304.

    MathSciNet  Article  Google Scholar 

  15. [15]

    V. G. Vizing: Coloring the vertices of a graph in prescribed colors (in Russian),Diskret. Analiz. 29,Metody Discret. Anal. v. Teorii Kodov i Shem 101 (1976), 3–10.

    MathSciNet  Google Scholar 

Download references

Author information



Additional information

Research supported in part by a United States-Israel BSF Grant and by a Bergmann Memorial Grant.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N., Tarsi, M. Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992).

Download citation

AMS Subject Classification codes (1991)

  • 05 C 15
  • 05 C 20