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

摘要

We prove that every Eulerian orientation of K m,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.