# Intersections of Cycling 2-factors

@article{Lipman2014IntersectionsOC, title={Intersections of Cycling 2-factors}, author={Drew J. Lipman}, journal={arXiv: Combinatorics}, year={2014} }

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…

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The proof gives an O(n2logn) time algorithm for finding an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings.