Z3-connectivity in Abelian Cayley graphs

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 G ′ such that for every map 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 total amount of f -values on the edges directed out from v minus the total amount of f -values on the edges directed into v equals b ( v ) . Jaeger et al. [F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combinatorial Theory, Series B 56 (1992) 165–182] conjectured that every 5-edge-connected graph G is Z 3 -connected, where Z 3 is the cyclic group of order 3. In this paper we prove that every connected Cayley graph G of degree at least 5 on an Abelian group is Z 3 -connected.