Non-zero disjoint cycles in highly connected group labeled graphs

Let G = ( V , E ) be an oriented graph whose edges are labeled by the elements of a group Γ. A cycle C in G has non-zero weight if for a given orientation of the cycle, when we add the labels of the forward directed edges and subtract the labels of the reverse directed edges, the total is non-zero. We are specifically interested in the maximum number of vertex disjoint non-zero cycles. ve that if G is a Γ-labelled graph and the corresponding undirected graph is 31 2 k -connected, either G has k disjoint non-zero cycles or it has a vertex set Q of order at most 2 k − 2 such that G − Q has no non-zero cycles. The bound “ 2 k − 2 ” is best possible.