Non-zero disjoint cycles in highly connected group labelled graphs

Let G = (V, E) be an oriented graph whose edges are labelled 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. We prove that if G is a Γ-labelled graph and G is the corresponding undirected graph, then if G is 31 2 k-connected, either G has k disjoint non-zero cycles or it has a vertex set Q of order at most 2k − 2 such that G − Q has no non-zero cycles. The bound “2k − 2” is best possible. This generalizes the results due to Thomassen [17], Rautenbach and Reed [12] and Kawarabayashi and Reed [9], respectively.