A note on internally disjoint alternating paths in bipartite graphs

Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x∈Xx∈X and y∈Yy∈Y. Let k be a positive integer. Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect to every perfect matching.