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We prove that every Eulerian orientation of Km,n contains<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V621MX58486572T3_html\26_2005_Article_0244_TeX2GIFIEq1.gif" border="0" alt="$\tfrac{1}{{4 + \sqrt 8 }}mn(1 - o(1))$" /> 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<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V621MX58486572T3_html\26_2005_Article_0244_TeX2GIFIEq2.gif" border="0" alt="$\tfrac{1}{{8 + \sqrt {32} }}n^2 (1 - o(1))$" /> 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.

Packing 4-Cycles in Eulerian and Bipartite Eulerian Tournaments with an Application to Distances in Interchange Graphs