Every N2-Locally Connected claw-Free Graph with Minimum Degree at Least 7 is Z3-Connected

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 function b : V (G) 7→ A satisfying ∑ v∈V (G) b(v) = 0, there is a function f : E(G) 7→ 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). Let Z3 denote the group of order 3. Jaeger et al conjectured that there exists an integer k such that every k-edge-connected graph is Z3-connected. In this paper, we prove that every N2-locally connected claw-free graph G with minimum degree δ(G) ≥ 7 is Z3connected.