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

Abstract

We prove that every Eulerian orientation of Km,n contains \(\tfrac{1} {{4 + \sqrt 8 }}mn(1 - o(1))\) 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 \(\tfrac{1} {{8 + \sqrt {32} }}n^2 (1 - o(1))\) 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.