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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 (fromto)  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 articlerefereed 
Keywords
 Clique
 Monotonic matrix
 Packing
 Semicross
 Stein corner
 Tripod
 52C17
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Dataset for New Results on Tripod Packings
Östergård, P. (Creator) & Pöllänen, A. (Creator), 26 Apr 2018
Dataset
Projects
 1 Finished

Construction and Classification of Discrete Mathematic Structures
Kokkala, J., Laaksonen, A., Heinlein, D., Ganzhinov, M., Östergård, P., Szollosi, F. & Pöllänen, A.
01/09/2015 → 31/08/2019
Project: Academy of Finland: Other research funding