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

[1]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

[2]  Paul D. Seymour,et al.  Nowhere-zero 6-flows , 1981, J. Comb. Theory, Ser. B.

[3]  G. Szekeres,et al.  Polyhedral decompositions of cubic graphs , 1973, Bulletin of the Australian Mathematical Society.

[4]  F. Jaeger,et al.  Flows and generalized coloring theorems in graphs , 1979, J. Comb. Theory, Ser. B.

[5]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[6]  Nathan Linial,et al.  Group connectivity of graphs - A nonhomogeneous analogue of nowhere-zero flow properties , 1992, J. Comb. Theory, Ser. B.

[7]  W. T. Tutte,et al.  On the Imbedding of Linear Graphs in Surfaces , 1949 .

[8]  Paul A. Catlin.  The reduction of graph families closed under contraction , 1996, Discret. Math..

[9]  Hong-Jian Lai.  Group Connectivity of 3-Edge-Connected Chordal Graphs , 2000, Graphs Comb..