Cycles of All Possible Lengths in Diregular Bipartite Tournaments

In 1967 Alspach [1] proved that every arc of a diregular tournament is contained in cycles of all possible lengths. In this paper, we extend this result to bipartite tournaments by showing that every arc of a diregular bipartite tournament is contained in cycles of all possible even lengths, unless it is isomorphic to one of the graphs \(F_{4k} \). Simultaneously, we also prove that an almost diregular bipartite tournament \(R\) is Hamiltonian if and only if \(|V_1| = |V_2|\) and \(R\) is not isomorphic to one of the graphs \(F_{4k-2}\), where \((V_1, V_2)\) is a bipartition of \(R\). Moreover, as a consequence of our first result, it follows that every diregular bipartite tournament of order \(p\) contains at least \(p/4\) distinct Hamiltonian cycles. The graphs \(F_r = (V, A)\), (\(r \geq 2\)) is a family of bipartite tournaments with \(V = \{v_1, v_2, \ldots, v_r\}\) and \(A = \{v_iv_j | j – i \equiv 1 \pmod{4}\}\).