Group Coloring and List Group Coloring Are Π2P-Complete

  • Daniel Král’
  • Pavel Nejedlý
Conference paper

DOI: 10.1007/978-3-540-28629-5_19

Volume 3153 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Král’ D., Nejedlý P. (2004) Group Coloring and List Group Coloring Are Π2P-Complete. In: Fiala J., Koubek V., Kratochvíl J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg

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.

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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