Degree condition and Z3Z3-connectivity

Let GG be a 2-edge-connected simple graph on n≥3n≥3 vertices and AA an abelian group with |A|≥3|A|≥3. If a graph G∗G∗ is obtained by repeatedly contracting nontrivial AA-connected subgraphs of GG until no such a subgraph left, we say GG can be AA-reduced to G∗G∗. Let G5G5 be the graph obtained from K4K4 by adding a new vertex vv and two edges joining vv to two distinct vertices of K4K4. In this paper, we prove that for every graph GG satisfying max{d(u),d(v)}≥n2 where uv∉E(G)uv∉E(G), GG is not Z3Z3-connected if and only if GG is isomorphic to one of twenty two graphs or GG can be Z3Z3-reduced to K3K3, K4K4 or K4− or G5G5. Our result generalizes the former results in [R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and Z3Z3-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.