A generalization of 0-sum flows in graphs

Let G be a graph and H be an abelian group. For every subset S⊆H a map ϕ:E(G)→S is called an S-flow. For a given S-flow of G, and every v∈V(G), define s(v)=∑uv∈E(G)ϕ(uv). Let k∈H. We say that a graph G admits a k-sum S-flow if there is an S-flow such that for each vertex . We prove that if G is a connected bipartite graph with two parts X={x1,…,xr}, Y={y1,…,ys} and are real numbers, then there is an R-flow such that s(xi)=ci and s(yj)=dj, for if and only if . Also, it is shown that if G is a connected non-bipartite graph and c1,…,cn are arbitrary integers, then there is a Z-flow such that s(vi)=ci, for i=1,…,n if and only if the number of odd ci is even.