Degree condition and [formula omitted]-connectivity

Abstract

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 G5 be the graph obtained from K4 by adding a new vertex v and two edges joining v to two distinct vertices of K4. In this paper, we prove that for every graph G satisfying max{d(u),d(v)}≥n2 where uv∉E(G), G is not Z3-connected if and only if G is isomorphic to one of twenty two graphs or G can be Z3-reduced to K3, K4 or K4− or G5. Our result generalizes the former results in [R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and Z3-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.