A note on the number of edges guaranteeing a C4 in Eulerian bipartite digraphs

Abstract

Let G be an Eulerian bipartite digraph with vertex partition sizes m, n. We prove the following Turán-type result: If e(G) > 2mn/3 then G contains a directed cycle of length at most 4. The result is sharp. We also show that if e(G) = 2mn/3 and no directed cycle of length at most 4 exists, then G must be biregular. We apply this result in order to obtain an improved upper bound for the diameter of interchange graphs.