Packing edge-disjoint triangles in regular and almost regular tournaments

For a tournament T , let ν3(T ) denote the maximum number of pairwise edge-disjoint triangles (directed cycles of length 3) in T . Let ν3(n) denote the minimum of ν3(T ) ranging over all regular tournaments with n vertices (n odd). We conjecture that ν3(n) = (1 + o(1))n /9 and prove that n 11.43 (1− o(1)) ≤ ν3(n) ≤ n 9 (1 + o(1)) improving upon the best known upper bound of n −1 8 and lower bound of n 11.5 (1 − o(1)). The result is generalized to tournaments where the indegree and outdegree at each vertex may differ by at most βn.