Packing non-zero A-paths in an undirected model of group labeled graphs

Let Gamma be an abelian group, and let gamma : E(G) -> Gamma be a function assigning values in Gamma to every edge of a graph G. For a subgraph H of G, let gamma (H) = Sigma(e is an element of E(H)) gamma(e). For a set A of vertices of G, an A-path is a path with both endpoints in A and otherwise disjoint from A. In this article. we show that either there exist k vertex disjoint A-paths P(1), P(2)...., P(k) such that gamma(P(i)) not equal 0 for all 1 <= i <= k, or there exists a set X of vertices such that G - X does not contain a non-zero A-path with vertical bar X vertical bar <= 50k(4). (C) 2009 Elsevier Inc. All rights reserved.