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Let D be any of the 10 digraphs obtained by orienting the edges of K4 - e. We establish necessary and sufficient conditions for the existence of a (K∗ n,D)-design for 8 of these digraphs. Partial results as well as some nonexistence results are established for the remaining 2 digraphs. © 2019 Ryan C. Bunge et al., published by Sciendo 2019.
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