Packing edge-disjoint triangles in regular and almost regular tournaments

Abstract

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))n2/9 and prove thatn211.43(1−o(1))≤ν3(n)≤n29(1+o(1)) improving upon the best known upper bound of n2−18 and lower bound of n211.5(1−o(1)). The result is generalized to tournaments where the indegree and outdegree at each vertex may differ by at most βn.