A note on the number of edges guaranteeing a \(C_4\) in Eulerian bipartite digraphs (Q5960796)

It is shown that if \(G\) is an Eulerian bipartite digraph with parts containing \(m\) and \(n\) vertices, then \(G\) contains a directed \(C_4\) if the number of edges \(e(G) > 2mn/3\). An example is given to show that the result is sharp, so that the Turán extremal number is \(2mn/3\), and it is proved that any extremal graph without a directed \(C_4\) must be a biregular bipartite digraph (any two vertices in the same part have the same indegree and outdegree). This extremal result has applications in determining the diameter in an interchange graph.