TY  - JOUR
ID  - Fortuna2013
LB  - Fortuna2013
AU  - Fortuna, Artur
AU  - Skupień, Zdzisław
TI  - Universal third parts of any complete 2-graph and none of DK<sub>5</sub>
ST  - Universal third parts of any complete 2-graph and none of DK<sub>5</sub>
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2013
VL  - 33
IS  - 4
SP  - 685
EP  - 696
DO  - http://dx.doi.org/10.7494/OpMath.2013.33.4.685
UR  - http://dx.doi.org/10.7494/OpMath.2013.33.4.685
AB  - 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.
KW  - decomposition
KW  - remainder
KW  - universal parts
KW  - isomorphic parts
SN  - 1232-9274
ER  -