A dual version of the Brooks group coloring theorem

Let G be a 2-edge-connected undirected graph, A be an (additive) Abelian group, and A = A − {0}. A graph G is A-connected if G has an orientation D(G) such that for every mapping b: V (G) → A satisfying  v∈V (G) b(v) = 0, there is a function f : E(G) → A ∗ such that for each vertex v ∈ V (G), the sum of f over the edges directed out from v minus the sum of f over the edges directed into v equals b(v). For a 2-edge-connected graph G, define Λg (G) = min{k: for any Abelian group A with |A| ≥ k, G is A-connected }. Let P denote a path in G, let βG(P) be the minimum length of a circuit containing P , and let βi(G) be the maximum of βG(P) over paths of length i in G. We show that Λg (G) ≤ βi(G) + 1 for any integer i > 0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [G. Fan, H.-J. Lai, R. Xu, C.-Q. Zhang, C. Zhou, Nowhere-zero 3-flows in triangularly connected graphs, Journal of Combinatorial Theory, Series B 98 (6) (2008) 1325–1336], among others. In this paper, we completely determine all graphs Gwith Λg (G) = β2(G) + 1. © 2012 Elsevier B.V. All rights reserved.