Group Coloring and List Group Coloring Are Π2P-Complete

  • Daniel Král’
  • Pavel Nejedlý
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

Abstract

A graph G is A-ℓ-choosable for an Abelian group A and an integer ℓ ≤ |A| if for each orientation of G, each edge-labeling ϕ: E(G)→ A and each list-assignment \(L: V(G)\to {A\choose\ell}\), there exists a vertex-coloring c: V(G)→ A with c(v)∈ L(v) for each vertex v and with \(c(v)-c(u)\not=\varphi(uv)\) for each oriented edge uv of G. We prove a dichotomy result on the computational complexity of this problem. In particular, we show that the problem is Π\(_{\rm 2}^{P}\)-complete if ℓ≥ 3 for any group A and it is polynomial-time solvable if ℓ=1,2. This also settles the complexity of group coloring for all Abelian groups.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Tarsi, M.: Colorings and Orientations of Graphs. Combinatorica 12, 125–134 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Du, D.-Z., Ko, K.-I.: Theory of Computational Complexity. John Wiley & Sons, New York (2000)MATHGoogle Scholar
  3. 3.
    Erdös, P., Rubin, A.L., Taylor, H.: Choosability in Graphs. Congress. Numer. 26, 122–157 (1980)Google Scholar
  4. 4.
    Gutner, S.: The Complexity of Planar Graph Choosability. Discrete Math. 159, 119–130 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group Connectivity of Graphs — A Non-homogeneous Analogue of Nowhere-zero Flow. J. Combin. Theory Ser. B 56, 165–182 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Král’, D., Pangrác, O., Voss, H.-J.: A Note on Group Colorings (submitted)Google Scholar
  7. 7.
    Lai, H.-J., Zhang, X.: Group Colorability of Graphs. Ars Combin. 62, 299–317 (2002)MATHMathSciNetGoogle Scholar
  8. 8.
    Lai, H.-J., Zhang, X.: Group Chromatic Number of Graphs without K5-minors. Graphs Comb. 18, 147–154 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lam, P.C.B.: The 4-choosability of Plane Graphs without 4-cycles. J. Combin. Theory Ser. B 76, 117–126 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  11. 11.
    Thomassen, C.: 3-list-coloring Planar Graphs of Girth 5. J. Combin. Theory Ser. B 64, 101–107 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Thomassen, C.: Every Planar Graph is 5-choosable. J. Combin. Theory Ser. B 62, 180–181 (1994)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Zhang, C.Q.: Integer Flows and Cycle Covers of Graphs. Marcel Dekker, New York (1996)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Daniel Král’
    • 1
  • Pavel Nejedlý
    • 1
  1. 1.Department of Applied Mathematics and, Institute for Theoretical Computer ScienceCharles UniversityPrague 1Czech Republic

Personalised recommendations