Group Coloring and List Group Coloring Are Π₂ ^P-Complete

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.