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

Let Γ be an abelian group, and let be a function assigning values in Γ to every edge of a graph G. For a subgraph H of G, let γ(H)=∑e∈E(H)γ(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 P1,P2,…,Pk such that γ(Pi)≠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 |X|⩽50k4.