Article

Periodica Mathematica Hungarica

, Volume 21, Issue 2, pp 95-100

First online:

A combinatorial approach for Keller's conjecture

  • K. CorrádiAffiliated withDept. of computer Techn, Eötvös Loránd univDept. of civil engineering math, Tech. univ. Budapest
  • , S. SzabóAffiliated withDept. of computer Techn, Eötvös Loránd univDept. of civil engineering math, Tech. univ. Budapest

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Abstract

The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a counterexample for this conjecture if and only if the following graphsΓ n has a 2 n size clique. The 4 n vertices ofΓ n aren-tuples of integers 0, 1, 2, and 3. A pair of thesen-tuples are adjacent if there is a position at which the difference of the corresponding components is 2 modulo 4 and if there is a further position at which the corresponding components are different. We will give the size of the maximal cliques ofΓ n forn≤5.

Mathematics Subject Classification 1980/85

Primary 10E30 Secondary 20K01

Key words and phrases

Cube tilings Keller's conjecture factorization of finite abelian groups