### Abstract

Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of n/2 independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d-dimensional cube for any fixed d ≥ 2. Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the l_{p} norm, for any fixed 1<p≤∞). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of Kn colours, where k=k(d) a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle.

Original language | English |
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Pages (from-to) | 587-606 |

Number of pages | 20 |

Journal | Random Structures and Algorithms |

Volume | 51 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2017 |

### Keywords

- Hamilton cycles
- perfect matchings
- rainbow
- random geometric graphs

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## Cite this

*Random Structures and Algorithms*,

*51*(4), 587-606. https://doi.org/10.1002/rsa.20717