Group Connectivity in Products of Graphs

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 � v∈V (G) b(v) = 0, 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}. Let G1 ⊗G2 and G1 ×G2 denote the strong and Cartesian product of two connected nontrivial graphs G1 and G2. In this paper, we prove that Λg(G1 ⊗G2) ≤ 4, where equality holds if and only if both G1 and G2 are trees and min{|V (G1)|, |V (G2)|}=2; Λg(G1 × G2) ≤ 5, where equality holds if and only if both G1 and G2 are trees and either G1 ∼ K1,m and