Orientations making kcycles cyclic
Abstract
We show that the minimum number of orientations of the edges of the nvertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n1)\rceil$. More generally, we also determine the minimum number of orientations of $K_n$ such that at least one of them orients some specific $k$cycles cyclically on every $k$element subset of the vertex set. The questions answered by these results were motivated by an analogous problem of Vera T. Sós concerning triangles and $3$edgecolorings. Some variants of the problem are also considered.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.06888
 Bibcode:
 2015arXiv150206888H
 Keywords:

 Mathematics  Combinatorics;
 05C35;
 05C20
 EPrint:
 9 pages