Contractible configurations on 3-flows in graphs satisfying the Fan-condition

Let G be a 2 -edge-connected graph of order n such that each pair of vertices x and y at distance 2 satisfies max { d ( x ) , d ( y ) } > n 2 (weak Fan-condition, [G.H. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984) 221–227]). It is obtained in this paper that G admits a nowhere-zero 3 -flow and is a 3 -flow contractible configuration except for a family of well-characterized counterexamples. This result generalizes some earlier results about 3 -flows, 3 -flow contractible configurations under some degree-sum conditions ([G. Fan, C. Zhou, Ore condition and nowhere-zero 3-flows, SIAM J. Discrete Math. 22 (2008) 288–294], [R. Luo, R. Xu, J.H. Yin, G.X. Yu, Ore-condition and Z 3 -connectivity, European J. Combin. 29 (2008) 1587–1595]).