Degree conditions for group connectivity

Let GG be a 2-edge-connected simple graph on n≥13n≥13 vertices and AA an (additive) abelian group with |A|≥4|A|≥4. In this paper, we prove that if for every uv∉E(G)uv∉E(G), max{d(u),d(v)}≥n/4max{d(u),d(v)}≥n/4, then either GG is AA-connected or GG can be reduced to one of K2,3,C4K2,3,C4 and C5C5 by repeatedly contracting proper AA-connected subgraphs, where CkCk is a cycle of length kk. We also show that the bound n≥13n≥13 is the best possible.