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 1186 Jin Yan, Senmei Yao, Hong-Jian Lai and Xiaofeng Gu G2 ∼= K1,n, for n, m ≥ 2 or min{|V (G1)|, |V (G2)|}=2. A similar result for the lexicographical product graphs is also obtained.