Complementary Cycles Containing a Fixed Arc and a Fixed Vertex in Bipartite Tournaments

Suppose that \(R = (V, A)\) is a diregular bipartite tournament of order \(p \geq 8\). Denote a cycle of length \(k\) by \(C_k\). Then for any \(e \in A(R)\), \(w \in V(R) \setminus V(e)\), there exists a pair of vertex-disjoint cycles \(C_4\) and \(C_{p-4}\) in \(R\) with \(e \in C_4\) and \(w \in C_{p-4}\), except \(R\) is isomorphic to a special digraph \(\tilde{F}_{4k}\).