Universal third parts of any complete 2-graph and none of DK5

Abstract

It is shown that there is no digraph $F$ which could decompose the complete digraph on 5 vertices minus any 2-arc remainder into three parts isomorphic to $F$ for each choice of the remainder. On the other hand, for each $n\ge3$ there is a universal third part $F$ of the complete 2-graph $^2K_n$ on $n$ vertices, i.e., for each edge subset $R$ of size $|R|=|^2K_n| \bmod 3$, there is an $F$-decomposition of $^2K_n-R$. Using an exhaustive computer-aided search, we find all, exactly six, mutually nonisomorphic universal third parts of the 5-vertex 2-graph. Nevertheless, none of their orientations is a universal third part of the corresponding complete digraph.