Naslov
Cube packings, second moment and holes

Autori
Dutour Sikirić, Mathieu ; Itoh, Yoshiaki ; Poyarkov, Alexei

Izvornik
European journal of combinatorics (0195-6698) 28 (2007); 715-725

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
cube ; tilings ; extremal problems.

Sažetak
We consider tilings and packings of R^d by integral translates of cubes [0, 2[^d, which are 4Z^d-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimensions d<=4. For higher dimensions, we use random methods for generating some examples. Such a cube packing is called non-extendible if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that d-dimensional cube packings with more than 2^d-3 cubes can be extended to cube tilings. We also give a lower bound on the number N of cubes of non-extendible cube packings. Given such a cube packing and z in Z^d, we denote by Nz the number of cubes inside the 4-cube z+[0, 4[^d and call second moment the average of Nz^2. We prove that the regular tiling by cubes has maximal second moment and give a lower bound on the second moment of a cube packing in terms of its density and dimension.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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