On group hypo-connected graphs

Let G be an undirected graph, A an abelian group and A* = A - {0}. A graph G is A-connected if G has an orientation D(G) such that for every map b : V(G) -> A satisfying Sigma(v is an element of v(G)) b(v) = 0, there is a function f : E(G) -> A, such that for each vertex v is an element of 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 is equal to b(v). For an edge e of G, by G/e we denote the contraction graph obtained from G by deleting the edge and identify its ends. A graph G is A-hypo-connected if G/e is A-connected for any edge e is an element of E(G). In this paper we give two kinds of characterization of the group hypo-connected graphs. For a Z(3)-hypo-connected but not Z(3)-connected graph G, we present the upper bounds on the minimum number of edges of G and lower bounds on the maximum number of edges of G, where Z(3) is the cyclic group of order three.