Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n ≥ 13 vertices and A an (additive) abelian group with | A | ≥ 4 . In this paper, we prove that if for every u v ∉ E ( G ) , max { d ( u ) , d ( v ) } ≥ n / 4 , then either G is A -connected or G can be reduced to one of K 2 , 3 , C 4 and C 5 by repeatedly contracting proper A -connected subgraphs, where C k is a cycle of length k . We also show that the bound n ≥ 13 is the best possible.