### Abstract

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 |

### Cite this

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

**On semi-simple drawings of the complete graph.** / Aichholzer, Oswin; Ebenführer, Florian; Parada, Irene; Pilz, Alexander; Vogtenhuber, Birgit.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

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

}

TY - GEN

T1 - On semi-simple drawings of the complete graph

AU - Aichholzer, Oswin

AU - Ebenführer, Florian

AU - Parada, Irene

AU - Pilz, Alexander

AU - Vogtenhuber, Birgit

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

M3 - Conference contribution

SP - 25

EP - 28

BT - Proc. XVII Encuentros de Geometría Computacional

CY - Alicante, Spain

ER -