In this paper, we consider the problem of decomposing the complete directed graph $K_n^*$ into cycles of given lengths. We consider general necessary conditions for a directed cycle decomposition of $K_n^*$ into $t$ cycles of lengths $m_1, m_2, \ldots, m_t$ to exist and and provide a powerful construction for creating such decompositions in the case where there is one 'large' cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case. In particular, when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.
@article{10_37236_8219,
author = {A. C. Burgess and P. Danziger and M. T. Javed},
title = {Cycle decompositions of complete digraphs},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {1},
doi = {10.37236/8219},
zbl = {1458.05211},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8219/}
}
TY - JOUR
AU - A. C. Burgess
AU - P. Danziger
AU - M. T. Javed
TI - Cycle decompositions of complete digraphs
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8219/
DO - 10.37236/8219
ID - 10_37236_8219
ER -
%0 Journal Article
%A A. C. Burgess
%A P. Danziger
%A M. T. Javed
%T Cycle decompositions of complete digraphs
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8219/
%R 10.37236/8219
%F 10_37236_8219
A. C. Burgess; P. Danziger; M. T. Javed. Cycle decompositions of complete digraphs. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/8219
