A Sufficient Condition on Group Connectivity of Graphs

Let A be an Abelian group, n >= 3 be an integer, and ex(n,A) be the maximum integer such that every n-vertex simple graph with at most ex(n,A) edges is not A-connected. In this paper, we obtain a necessary condition for a graph being A-connected. Employing the condition we present a lower bound for ex(n, Z(3)) which improves some known result and prove that every cubic graph (not necessarily simple graph) with order at least 18 is not Z(3)-connected.