Orthogonal representations over finite fields and the chromatic number of graphs

Abstract

We study the relationship between the minimum dimension of an orthogonal representation of a graph over a finite field and the chromatic number of its complement. It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not. This implies the NP-hardness of determining the minimum rank of a matrix in such a class. Finally we give for any class of matrices defined by a graph that is interesting in this respect a reduction of the 3-colorability problem to the problem of deciding whether or not this class contains a matrix of rank equal to three.

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

References

  1. [1]

    R. L. Brooks: On colouring the nodes of a network,Proceedings of the Cambridge Philosophical Society,37 (1941), 194–197.

    Google Scholar 

  2. [2]

    A. E. Brouwer, A. M. Cohen, andA. Neumaier:Distance-Regular Graphs, Ergebnisse der Mathematik 3.18, Springer, Heidelberg (1989).

    Google Scholar 

  3. [3]

    S. Even:Graph Algorithms, Pitman, (1979).

  4. [4]

    M. R. Garey, D. S. Johnson, andL. J. Stockmeyer: Some NP-Complete Graph Problems,Theor. Comput. Sci.,1 (1976), 237–267.

    Google Scholar 

  5. [5]

    M. R. Garey, andD. S. Johnson: The Complexity of Near-Optimal Graph Coloring,J. ACM.,23 (1976), 43–49.

    Google Scholar 

  6. [6]

    M. R. Garey, andD. S. Johnson:Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, (1979).

    Google Scholar 

  7. [7]

    M. Grötchel, L. Lovász, andA. Schrijver: The ellipsoid method and its consequences in combinatorial optimization,Combinatorica,1 (2) (1981), 169–197.

    Google Scholar 

  8. [8]

    W. Haemers: On the problems of Lovász concerning the Shannon capacity of a graph,IEEE Trans. Inform. Theory,25 (1979), 231–232.

    Google Scholar 

  9. [9]

    W. Haemers: An upper bound for the Shannon capacity of a graph,Colloqua Mathematica Societatis János Bolyai,25, Algebraic Methods in Graph Theory, Szeged (Hungary), (1978), 267–272.

  10. [10]

    D. Hershkowitz, andH. Schneider: Ranks of zero patterns and sign patterns, preprint, (1991).

  11. [11]

    A. J. Hoffman: On eigenvalues and colorings of graphs, in B. Harris, Ed.,Graph Theory and its applications, New York and London: Academic, (1970), 79–91.

    Google Scholar 

  12. [12]

    D. E. Knuth: The Sandwich Theorem,Electronic J. Comb.,1 (1994), #A1.

    Google Scholar 

  13. [13]

    L. Lovász: On the Shannon capacity of a graph,IEEE Trans. Inform. Theory,25 (1979), 1–7.

    Google Scholar 

  14. [14]

    L. Lovász: An Algorithmic Theory of Numbers, Graphs and Convexity,CBMS Regional Conference Series in Applied Mathematics, SIAM, (1986) §3.2.

  15. [15]

    L. Lovász, M. Saks, andA. Schrijver: Orthogonal Representations and Connectivity of Graphs,Linear Algebra Appl.,114/115 (1989), 439–454.

    Google Scholar 

  16. [16]

    C. H. Papadimitriou:Computational Complexity, Addison-Wesley Publ. Co., (1994).

  17. [17]

    R. Peeters:Ranks and Structure of Graphs, dissertation, Tilburg University, (1995).

  18. [18]

    A. A. Razborov: Applications of matrix methods to the theory of lower bounds in computational complexity,Combinatorica,10 (1) (1990), 81–93.

    Google Scholar 

  19. [19]

    A. Schrijver: A comparison of the Delsarte and Lovász bounds,IEEE Trans. Inform. Theory,25 (1979), 425–429.

    Google Scholar 

  20. [20]

    C. E. Shannon: The zero-error capacity of a noisy channel,IRE Trans. Inform. Theory,3 (1956), 3–15.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Peeters, R. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica 16, 417–431 (1996). https://doi.org/10.1007/BF01261326

Download citation

Mathematics Subject Classification (1991)

  • 05C15
  • 05C20