Decomposing complete graphs into cubes

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 su‐cient for d equal to 3 or 5. For larger values of d, the necessary conditions are asymptotically su‐cient by Wilson’s results. We prove that for each odd d there is an inflnite arithmetic progression of even integers n for which a decomposition exists. This lends further weight to a long-standing conjecture of Kotzig.