Orientations Making <Emphasis Type="Italic">k</Emphasis>-Cycles Cyclic

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is \(\lceil \log _2(n-1)\rceil \). 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. Though only formally related, the questions answered by these results were motivated by an analogous problem of Vera T. Sós concerning triangles and 3-edge-colorings. Some variants of the problem are also considered.