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

Let Γ be an abelian group, and let γ : E ( G ) → Γ 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 P 1 , P 2 , … , P k such that γ ( P i ) ≠ 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 | ⩽ 50 k 4 .