A dual version of the Brooks group coloring theorem

Abstract

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 G with Λg(G)=β2(G)+1.