### Abstract

For n even, a factorization of a complete graph K_{n} is a partition of the edges into n-1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizations have orthogonal paths. Assisted by a simple edge-switching algorithm, here we show that for n≥8, the rotational factorization of K_{n}, GK_{n} has orthogonal paths. We prove that this algorithm finds a rainbow path with at least (2n+1)/3 vertices in any factorization of K_{n} (in fact, in any proper coloring of K _{n}). We also give some problems and conjectures about the properties of the algorithm.

Original language | English |
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Pages (from-to) | 167-176 |

Number of pages | 10 |

Journal | Journal of Combinatorial Designs |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2010 |

### Keywords

- Factorizations of complete graphs
- Rainbow and orthogonal paths

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

_{n}.

*Journal of Combinatorial Designs*,

*18*(3), 167-176. https://doi.org/10.1002/jcd.20243