@Article{Fortuna2013,
author = "Fortuna, Artur and Skupień, Zdzisław",
title = "Universal third parts of any complete 2-graph and none of DK<sub>5</sub>",
journal = "Opuscula Math.",
year = "2013",
volume = "33",
number = "4",
pages = "685--696",
doi = "http://dx.doi.org/10.7494/OpMath.2013.33.4.685",
url = "http://dx.doi.org/10.7494/OpMath.2013.33.4.685",
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.",
keywords = "decomposition, remainder, universal parts, isomorphic parts",
issn = "1232-9274"
}