St Reading Discrete Mathematics, Algorithms and Applications Every N 2 -locally Connected Claw-free Graph with Minimum Degree at Least 7 Dedicated to the 70th Birthday of Professor Feng Tian

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and 19 A∗ = A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every 20 function b : V (G) → A satisfying Pv∈V (G) b(v) = 0, there is a function f : E(G) → A∗ 21 such that for each vertex v ∈ V (G), the total amount of f values on the edges directed 22 out from v minus the total amount of f values on the edges directed into v equals 23 b(v). Let Z3 denote the group of order 3. Jaeger et al. conjectured that there exists 24 an integer k such that every k-edge-connected graph is Z3-connected. In this paper, we 25 prove that every N2-locally connected claw-free graph G with minimum degree δ(G) ≥ 7 26 is Z3-connected. 27