Abstract
For the complete digraph DK n 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 DK 5 in four of the six possible cases. For n=9+6k, k=0,1,…, a third part D of DK n is found such that D is a self-converse oriented graph and all D-decompositions of DK n have trivial automorphism group.
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