On the complexity of graph critical uncolorability
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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- [AH]K. Appel and W. Haken, “Every Planar Graph is 4-Colorable”, Ill. J. Math 21, 429–567 Google Scholar
- [B]R. Brooks, “On Coloring the Nodes of a Network”, Proc. Combridge Philos. Soc. 37, 194–197. Google Scholar
- [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. Google Scholar
- [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
- [GJ]M.Garey and D.Johnson, Computers and Intractability. W.H.Freeman and Company. Google Scholar
- [K]R. Karp, “Reducibility Among Combinatorial Problems”, in R.Miller and J.Thatcher (eds.), Complexity of Computer Computations, Plenum Press. 85–103. Google Scholar
- [O]O. Ore, “Four Color Conjecture.” Academic Press, .Google Scholar
- [PW]C.H.Papadimitriou and D. Wolfe, “The Complexity of Facet Resolved”, Proc. of IEEE Foundations on Computer Science. .Google Scholar
- [PY]C.H. Papadimitriou and M. Yannakakis, “The Complexity of Facets (and Some Facets of Complexity)”, JCSS 28,2,244–259Google Scholar
- [S]L. Stockmeyer, “Planar 3-Colorability is NP-Complete”, SIGACT News,5:3 19–25.Google Scholar