Orientations making k-cycles cyclic

TL;DR

This paper determines the minimum number of orientations needed for a complete graph to ensure every triangle or specific k-cycles are cyclic in at least one orientation, extending to general k-cycles and motivated by related coloring problems.

Contribution

It provides exact values for the minimum orientations required to make all triangles or k-cycles cyclic, generalizing previous triangle-focused results and exploring variants.

Findings

Minimum orientations for triangles is eil(log_2(n-1)).

Extended results for k-cycles and specific subsets.

Connections to Vera Sf3s's triangle coloring problem.

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.