A dual version of Brook’s coloring theorem

LetG = (V,E) be a graph with a fixed orientation and let A be an abelian group. Let F (G,A) denote the set of all functions from E(G) to A. The graph G is A-connected if for any function f̄ ∈ F (G,A), there exists an A-flow f such that f(e) 6= f̄(e) for any e ∈ E(G). The graph G is A-colorable if for any function f ∈ F (G,A), there exists a function c : V (G) → A such that for any arc e = (u, v) in G, c(u) − c(v) 6= f(e). The group connectivity number Λg(G) of a graph G is the minimum k such that G is A-connected for any group A of order at least k. The group chromatic number χg(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m. The group connectivity number and the group chromatic number are dual concepts. Brook’s theorem states that χ(G) ≤ ∆(G) + 1, where equality holds if and only if G is an odd cycle or a complete graph. In [Ars Combinatoria, 62 (2002), 299-317], Lai et al. proved the group analogue which states that χg(G) ≤ ∆(G) + 1, where equality holds if and only if G is a cycle or a complete graph. Let P2 denote a path with 2 edges. Define g2(P2) = min{|C|: C is a cycle containing P2} and g2(G) = max{g2(P2): P2 ⊂ G}. In this paper we prove a dual version of Brook’s theorem that Λg(G) ≤ g2(G) + 1 and also characterize the extremal graphs.