For the complete digraph script script D signKn with n≥3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph script D signK5 in four of the six possible cases. For n = 9 + 6k, k = 0, 1,..., a third part D of script D signKn is found such that D is a self-converse oriented graph and all D-decompositions of script D signKn have trivial automorphism group. © 2000 Elsevier Science B.V. All rights reserved.
Meszka, M., & Skupień, Z. (2000). On some third parts of nearly complete digraphs. Discrete Mathematics, 212(1–2), 129–139. https://doi.org/10.1016/S0012-365X(99)00214-9