Nowhere-zero 3-flows in locally connected graphs

Let G be a graph. For each vertex v 2V (G), Nv denotes the subgraph induces by the vertices adjacent to v inG. The graphG is locally k edge-connected if for each vertex v 2V (G), Nv is k-edge-connected. In this paper we study the existence of nowhere-zero 3-flows in locally k-edgeconnected graphs. In particular, we show that every 2-edge-connected, locally 3-edge-connected graph admits a nowhere-zero 3-flow. This result is best possible in the sense that there exists an infinite family of 2-edgeconnected, locally 2-edge-connected graphs each of which does not have a 3-NZF. 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003