Packing 4-cycles in Eulerian and bipartite Eulerian tournaments with an application to distances in interchange graphs

We prove that every Eulerian orientation of $K_{m,n}$ contains $\frac{1}{4+\sqrt{8}}mn(1-o(1))$ arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with $n$ vertices contains $\frac{1}{8+\sqrt{32}}n^2(1-o(1))$ arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between two antipodal vertices in interchange graphs.