On the complexity of graph critical uncolorability

  • Jin-yi Cai
  • Gabriele E. Meyer
Algorithms And Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)


In their paper, C.H.Papadimitriou and M.Yannakakis [PY] posed the problem of classifying the complexity of graph critical uncolorability; and in particular, they asked whether the minimal-3-uncolorability problem is DP-complete. This paper gives an affirmative answer to the above question. We show that minimal-k-uncolorability is DP-complete, for all fixed k≥3. Furthermore, the reduction can be modified by using “sensitive” transformations to resolve the planar case (for k=3), bounded vertex degree case and their combination.


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  1. [AH]
    K. Appel and W. Haken, “Every Planar Graph is 4-Colorable”, Ill. J. Math 21, 429–567 [1977]Google Scholar
  2. [B]
    R. Brooks, “On Coloring the Nodes of a Network”, Proc. Combridge Philos. Soc. 37, 194–197. [1941]Google Scholar
  3. [BG]
    A. Blass and Y. Gurevich, “On the Unique Satisfiability Problem”, Information and Control 55, 80–88. [1982]CrossRefGoogle Scholar
  4. [CC]
    T. Coleman and J. Cai, “The Cyclic Coloring Problems and Estimation of Sparse Hessian Matrices”, SIAM Journal on Algebraic and Discrete Methods. Vol 7, pp 221–235. [1986]Google Scholar
  5. [CH]
    J. Cai and L. Hemachandra, “The Boolean Hierarchy: Hardware over NP”, Proc. of the Structure in Complexity Theory Conference, Springer-Verlag Lecture Notes in Computer Science 1986.Google Scholar
  6. [GJ]
    M.Garey and D.Johnson, Computers and Intractability. W.H.Freeman and Company. [1979]Google Scholar
  7. [GJS]
    M. Garey, D. Johnson and L. Stockmeyer, “Some Simplified NP-Complete Graph Problems”, Theoretical Computer Science 1, 237–267.[1976]CrossRefGoogle Scholar
  8. [K]
    R. Karp, “Reducibility Among Combinatorial Problems”, in R.Miller and J.Thatcher (eds.), Complexity of Computer Computations, Plenum Press. 85–103. [1972]Google Scholar
  9. [O]
    O. Ore, “Four Color Conjecture.” Academic Press, [1976].Google Scholar
  10. [PW]
    C.H.Papadimitriou and D. Wolfe, “The Complexity of Facet Resolved”, Proc. of IEEE Foundations on Computer Science. [1985].Google Scholar
  11. [PY]
    C.H. Papadimitriou and M. Yannakakis, “The Complexity of Facets (and Some Facets of Complexity)”, JCSS 28,2,244–259[1982]Google Scholar
  12. [S]
    L. Stockmeyer, “Planar 3-Colorability is NP-Complete”, SIGACT News,5:3 19–25.[1973]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Jin-yi Cai
    • 1
  • Gabriele E. Meyer
    • 2
  1. 1.Department of Computer ScienceYale UniversityNew Haven
  2. 2.Mathematics DepartmentCornell UniversityIthaca

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