Geodesic
Decomposing complete graphs into cubes
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 141-147

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This paper concerns when the complete graph on n vertices can be decomposed into d-dimensional cubes, where d is odd and n is even. (All other cases have been settled.) Necessary conditions are that n be congruent to 1 modulo d and 0 modulo 2^d. These are known to be sufficient for d equal to 3 or 5. For larger values of d, the necessary conditions are asymptotically sufficient by Wilson's results. We prove that for each odd d there is an infinite arithmetic progression of even integers n for which a decomposition exists. This lends further weight to a long-standing conjecture of Kotzig.
Keywords: graph decomposition, graph factorization, d-cube 
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El-Zanati, Saad; Eynden, C. Decomposing complete graphs into cubes. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 141-147. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a12/