### Abstract

In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor n2 rfloor-2)$-bishellability also guarantees, with a simpler proof than for $s$-shellability, that a drawing has at least $H(n)$ crossings. We exhibit a drawing of $K_11$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved.

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

Place of Publication | Alicante, Spain |

Pages | 17-20 |

Number of pages | 4 |

Publication status | Published - 2017 |

## Cite this

Ábrego, B. M., Aichholzer, O., Fernández-Merchant, S., McQuillan, D., Mohar, B., Mutzel, P., ... Vogtenhuber, B. (2017). Bishellable drawings of $K_n$. In

*Proc. XVII Encuentros de Geometría Computacional*(pp. 17-20). Alicante, Spain.