### Abstract

In this work we study rotation systems and semi-simple drawings of $K_n$. A simple drawing of a graph is a drawing in which every pair of edges intersects in at most one point. In a semi-simple drawing, edge pairs might intersect in multiple points, but incident edges only intersect in their common endpoint. A rotation system is called (semi-)realizable if it can be realized with a (semi-)simple drawing. It is known that a rotation system is realizable if and only if all its 5-tuples are realizable. For the problem of characterizing semi-realizability, we present a rotation system with six vertices that is not semi-realizable, although all its 5-tuples are semi-realizable. Moreover, by an exhaustive computer search, we show that also for seven vertices there exist minimal not semi-realizable rotation systems (that is, rotation systems in which all proper sub-rotation systems are semi-realizable). This indicates that checking semi-realizability is harder than checking realizability. Finally we show that for semi-simple drawings, generalizations of Conway's Thrackle Conjecture and the conjecture on the existence of plane Hamiltonian cycles do not hold.

Original language | English |
---|---|

Title of host publication | Proc. XVII Encuentros de Geometría Computacional |

Place of Publication | Alicante, Spain |

Pages | 25-28 |

Number of pages | 4 |

Publication status | Published - 2017 |

Event | XVII Spanish Meeting on Computational Geometry - , Spain Duration: 26 Jun 2017 → 28 Jun 2017 https://dmat.ua.es/en/egc17/documentos/book-of-abstracts.pdf |

### Conference

Conference | XVII Spanish Meeting on Computational Geometry |
---|---|

Abbreviated title | EGC 2017 |

Country | Spain |

Period | 26/06/17 → 28/06/17 |

Internet address |

## Cite this

Aichholzer, O., Ebenführer, F., Parada, I., Pilz, A., & Vogtenhuber, B. (2017). On semi-simple drawings of the complete graph. In

*Proc. XVII Encuentros de Geometría Computacional*(pp. 25-28). Alicante, Spain.