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### Abstract

Consider an n× n× n cube Q consisting of n
^{3} unit cubes. A tripod of order n is obtained by taking the 3 n- 2 unit cubes along three mutually adjacent edges of Q. The unit cube corresponding to the vertex of Q where the edges meet is called the center cube of the tripod. The function f(n) is defined as the largest number of integral translates of such a tripod that have disjoint interiors and whose center cubes coincide with unit cubes of Q. The value of f(n) has earlier been determined for n≤ 9. The function f(n) is here studied in the framework of the maximum clique problem, and the values f(10) = 32 and f(11) = 38 are obtained computationally. Moreover, by prescribing symmetries, constructive lower bounds on f(n) are obtained for n≤ 26. A conjecture that f(n) is always attained by a packing with a symmetry of order 3 that rotates Q around the axis through two opposite vertices is disproved.

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

Pages (from-to) | 271–284 |

Number of pages | 14 |

Journal | Discrete and Computational Geometry |

Volume | 61 |

Issue number | 2 |

Early online date | 18 Jun 2018 |

DOIs | |

Publication status | Published - 15 Mar 2019 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Clique
- Monotonic matrix
- Packing
- Semicross
- Stein corner
- Tripod
- 52C17

## Fingerprint Dive into the research topics of 'New Results on Tripod Packings'. Together they form a unique fingerprint.

## Projects

- 1 Finished

## Construction and Classification of Discrete Mathematic Structures

Kokkala, J., Laaksonen, A., Heinlein, D., Ganzhinov, M., Östergård, P. & Szollosi, F.

01/09/2015 → 24/09/2019

Project: Academy of Finland: Other research funding

## Cite this

*Discrete and Computational Geometry*,

*61*(2), 271–284. https://doi.org/10.1007/s00454-018-0012-2