Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈ V ( G ), N v denotes the subgraph induces by the vertices adjacent to v in G . The graph G is locally k ‐edge‐connected if for each vertex v ∈ V ( G ), N v is k ‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k ‐edge‐connected 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‐edge‐connected, 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