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Discussiones Mathematicae Graph Theory 26(1) (2006) 141-147
DOI: https://doi.org/10.7151/dmgt.1308

DECOMPOSING COMPLETE GRAPHS INTO CUBES

Saad I. El-Zanati and C. Vanden Eynden

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 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.

2000 Mathematics Subject Classification: 05C70, 05B30.

References

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Received 5 May 2005
Revised 19 September 2005