Degree condition and -connectivity

Let G be a 2-edge-connected simple graph on n ≥ 3 vertices and A an abelian group with | A | ≥ 3 . If a graph G ∗ is obtained by repeatedly contracting nontrivial A -connected subgraphs of G until no such a subgraph left, we say G can be A -reduced to G ∗ . Let G 5 be the graph obtained from K 4 by adding a new vertex v and two edges joining v to two distinct vertices of K 4 . In this paper, we prove that for every graph G satisfying max { d ( u ) , d ( v ) } ≥ n 2 where u v ∉ E ( G ) , G is not Z 3 -connected if and only if G is isomorphic to one of twenty two graphs or G can be Z 3 -reduced to K 3 , K 4 or K 4 − or G 5 . Our result generalizes the former results in [R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and Z 3 -connectivity, European J. Combin. 29 (2008) 1587–1595] by Luo et al., and in [G. Fan, C. Zhou, Ore condition and nowhere zero 3-flows, SIAM J. Discrete Math. 22 (2008) 288–294] by Fan and Zhou.