A generalization of 0-sum flows in graphs

Let G be a graph and H be an abelian group. For every subset S subset of H a map phi: E(G) -> S is called an S-flow. For a given S-flow of G, and every v is an element of V (G), define s(v) = Sigma(uv is an element of E(G)) phi(uv). Let k is an element of H. We say that a graph G admits a k-sum S-flow if there is an S-flow such that for each vertex v, s(v) = k. We prove that if G is a connected bipartite graph with two parts X = {x(1),...,x(r)}, Y = {y(1),...,y(s)} and c(1),..., c(r), d(1),...d(s) are real numbers, then there is an R-flow such that s(x(i)) = c(1) and s(y(j)) = d(j), for 1 <= i <= r, 1 <= j <= s if and only if Sigma(r)(i=1) c(i) = Sigma(s)(j=1) d(j). Also, it is shown that if G is a connected non-bipartite graph and c(1),..., c(n) are arbitrary integers, then there is a Z-flow such that s(v(i)) = c(i), for i = 1,..., n if and only if the number of odd c(i) is even. (C) 2013 Elsevier Inc. All rights reserved.