Group connectivity for Km,n-k

Abstract

Let G be an undirected graph, A be an (additive) abelian group with identity 0, and A* = A - {0}. A graph G is A-connected if G has an orientation such that for every function b : V(G) → A satisfying Sigma;_vεV(G) b( v) = 0, there is a function f : E(G) → A* such that at each vertex v ε V(G), the net flow out of v equals b(v). For a graph G, the group connectivity number of G is defined as follows: Λ_g(G) = min{h|G is A-connected for every abelian group A with |A| ≥ h}. In this paper, we determine the value of Λ_g(K_{m, n}^-k) for m ≥ n ≥ 3 except for the graph K_5,4^-4, where K_m,n^-k is the graph obtained from the m × n complete bipartite graph K_m,n by removing k independent edges.