Legal coloring of graphs


The following computational problem was initiated by Manber and Tompa (22nd FOCS Conference, 1981): Given a graphG=(V, E) and a real functionf:VR which is a proposed vertex coloring. Decide whetherf is a proper vertex coloring ofG. The elementary steps are taken to be linear comparisons.

Lower bounds on the complexity of this problem are derived using the chromatic polynomial ofG. It is shown how geometric parameters of a space partition associated withG influence the complexity of this problem.

Existing methods for analyzing such space partitions are suggested as a powerful tool for establishing lower bounds for a variety of computational problems.

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


  1. [1]

    D. P. Dobkin andR. J. Lipton, On the complexity of computations under varying sets of primitives,J. Computer and System Science,18 (1979), 86–91.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    C. Greene, Acyclic orientations (Notes),in: Higher Combinatorics (M. Aigner, ed.), pp. D. Reidel, Dordrecht, 1977, 65–68.

  3. [3]

    B. Grünbaum,Convex Polytopes, Interscience, New York, 1967.

    MATH  Google Scholar 

  4. [4]

    R. L. Graham, A. C. Yao andF. F. Yao, Information bounds are weak in the shortest distance problem,J. Assoc. Comp. Mach.,27 (1980), 428–444.

    MATH  MathSciNet  Google Scholar 

  5. [5]

    U. Manber andM. Tompa, The effect of number of Hamiltonian paths on the complexity of a vertex coloring problem.Proc. 22nd Annual Symposium on the Foundations of Computer Science, 1981.

  6. [6]

    R. Rivest andA. C. Yao, On the polyhedral decision problem,SIAM J. Computing,9 (1980), 343–347.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    M. Snir, Proving lower bounds for linear decision trees,in: Proc. 8th. International Coll. on Automata Languages and Programming, Lecture Notes in Computer Science 115, Springer Verlag, (1981), 305–315.

  8. [8]

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

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    T. Zaslavsky,Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Mem. AMS. 154, Amer. Math. Soc., Providence, R. I., 1975.

    Google Scholar 

  10. [10]

    T. Zaslavsky, The geometry of root systems and signed graphs,Amer. Math. Monthly (1981), 88–105.

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Linial, N. Legal coloring of graphs. Combinatorica 6, 49–54 (1986).

Download citation

AMS subject classification (1980)

  • 68 C 25
  • 05 C 15