Orientations making k-cycles cyclic

Abstract

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. The questions answered by these results were motivated by an analogous problem of Vera T. S\'os concerning triangles and $3$-edge-colorings. Some variants of the problem are also considered.