Group connectivity of complementary graphs

Abstract Let G be a 2‐edge‐connected undirected graph, A be an (additive) abelian group and A* = A−{0}. A graph G is A‐connected if G has an orientation D(G) such that for every function b: V(G)↦A satisfying , there is a function f: E(G)↦A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2‐edge‐connected graph G, define Λg(G) = min{k: for any abelian group A with |A|⩾k, G is A‐connected }. In this article, we prove the following Ramsey type results on group connectivity: Let G be a simple graph on n⩾6 vertices. If min{δ(G), δ(Gc)}⩾2, then either Λg(G)⩽4, or Λg(Gc)⩽4. Let Z3 denote the cyclic group of order 3, and G be a simple graph on n⩾44 vertices. If min{δ(G), δ(Gc)}⩾4, then either G is Z3‐connected, or Gc is Z3‐connected. © 2011 Wiley Periodicals, Inc. J Graph Theory