Intersections of Cycling 2-factors

Define an embedding of graph $G=(V,E)$ with $V$ a finite set of distinct points on the unit circle and $E$ the set of line segments connecting the points. Let $V_1,\ldots,V_k$ be a labeled partition of $V$ into equal parts. A 2-factor is said to be {\em cycling} if for each $u\in V$, $u\in V_i$ implies $u$ is adjacent to a vertex in $V_{i+1\: (mod \: k)}$ and a vertex in $V_{i-1\: (mod\: k)}$. In this paper, we will present some new results about cycling 2-factors including a tight upper bound on the minimum number of intersections of a cycling 2-factor for $k=3$.