K. Corradi and S. Szabo. A combinatorial approach for keller's conjecture. Periodica Mathematica Hungarica, pages 95--100, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in a lattice tiling of R n with unit n cubes, there must exist two cubes that share an (n 1) dimensional face. Minkowski s conjecture was proved by Hajos [22] in 1942. In 1930, Keller generalized the conjecture, suggesting that it holds even without the lattice assumption. Corradi and Szabo [9] proved that there is a counterexample to Keller s conjecture, if and only if, the graph G n with 4 n vertices has a maximum clique of size 2 n . The graph G n is defined as the graph with vertex set V of n tuples of integers 0, 1, 2 and 3, i.e. V n = d 1 , d 2 , d n ) d i # ....

K. Corradi and S. Szabo. A combinatorial approach for keller's conjecture. Periodica Mathematica Hungarica, pages 95--100, 1990.

....Perron [10] proved Keller s conjecture is true in dimensions n up to 6. In 1942 Hajos [2] proved that it holds for all lattice tilings of R n , which settled Minkowski s conjecture. Subsequently various reductions of Keller s conjecture were made in Hajos [3] Szabo [12] and Corradi and Szabo [1]. Recently, we showed that Keller s conjecture is false in all dimensions n 10, by construction, cf. Lagarias and Shor [6] It remains open in dimensions 7, 8 and 9. Let K n denote the largest integer such that every tiling of R n by unit cubes contains two cubes that have a common face of ....

K. Corradi and S. Szabo, A combinatorial approach for Keller's conjecture, Period. Math. Hungar. 21 (1990), 91-100.

.... ; d 2 ; d n ) d i 2 f0; 1; 2; 3g; i = 1; 2; ng where two vertices u = d 1 ; d 2 ; d n ) and v = d 0 1 ; d 0 2 ; d 0 n ) in V n are adjacent if and only if 9 i; 1 i n : d i Gamma d 0 i j 2 mod 4 (29) and 9 j 6= i; 1 j n : d j 6= d 0 j : 30) In [97], Corr adi and Szab o presented a graph theoretic equivalent of Keller s conjecture. It is shown that, there is a counterexample to Keller s conjecture if and only if there exist a n 2 N such that Gamma n has a clique of size 2 n . Gamma n has 4 n vertices, 1 2 4 n (4 n Gamma 3 n ....

K. Corradi and S. Szabo, A combinatorial approach for Keller's conjecture, Periodica Mathematica Hungarica, Vol. 21: 95--100, 1990.

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K. Corradi and S. Szabo, A Combinatorial Approach for Keller's Conjecture, Periodica Mathematica Hungarica, Vol. 21, No. 2: 95-100, 1990.

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