A combinatorial approach for Keller's conjecture
- Cite this article as:
- Corrádi, K. & Szabó, S. Period Math Hung (1990) 21: 95. doi:10.1007/BF01946848
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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 graphsΓn has a 2n size clique. The 4n vertices ofΓn 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 ofΓn forn≤5.