EVERY N2-LOCALLY CONNECTED CLAW-FREE GRAPH WITH MINIMUM DEGREE AT LEAST 7 IS Z3-CONNECTED

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 function b : V(G) ↦ A satisfying Συ∈V(G)b(υ) = 0, there is a function f : E(G) ↦ A* such that for each vertex υ ∈ V(G), the total amount of f values on the edges directed out from υ minus the total amount of f values on the edges directed into υ equals b(υ). Let Z3denote 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 Z3-connected.