ArticleOriginal scientific text
Title
Decomposing complete graphs into cubes
Authors 1, 1
Affiliations
- 4520 Mathematics Department, Illinois State University, Normal, Illinois 61790-4520, USA
Abstract
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 . 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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