## Abstract

In 1958, Hill conjectured that the minimum number of crossings in a drawing of K_{n} is exactly (Formula presented.). Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by x-monotone curves. In fact, our proof shows that the conjecture remains true for x-monotone drawings of K_{n} in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by Ábrego et al. We also give a combinatorial characterization of several classes of x-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Carathéodory’s theorem to simple drawings of complete graphs.

Original language | English |
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Pages (from-to) | 107-143 |

Number of pages | 37 |

Journal | Discrete and Computational Geometry |

Volume | 53 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |

## Keywords

- Complete graph
- Crossing number
- Monotone drawing
- Monotone odd crossing number
- Odd crossing number
- Shellable drawing

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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