Cubes
From Gdesigns
Relevant articles: _{[1]}, _{[2]}, _{[3]}, _{[4]}, _{[5]}, _{[6]}, _{[7]}, _{[8]}.
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Cubes
The cube or cube of dimension , denoted by , is the graph with vertex set consisting of all binary strings of length and with two vertices adjacent if and only if they differ in exactly one coordinate.
Spectrum Results
Table 1 summarises the known results on the spectrum for the cube. An explanation of the sources of these results is given in _{[1]}.
Table 1
Spectrum for the cube  Possible exceptions  

Theorem 1 _{[5]}
Let be odd, let be such that and let be the order of . If is a nonnegative integer and , then there exists a cubedesign of order .
Theorem 2 _{[4]}
Let be odd and let be such that . If is a nonnegative integer and , then there exists a cubedesign of order .
Notes
 The cube is a single edge and the spectrum is trivially the set of all positive integers.
 The cube is a cycle and the spectrum is well known to be all (see the section on cycles).
References
 ↑ ^{1.0} ^{1.1} Adams, P., Bryant, D., and Buchanan, M. A survey on the existence of Gdesigns, J. Combin. Des. 16, 373–410 (2008).
 ↑ Bryant, D. E., ElZanati, S. I., and Gardner, R. B. Decompositions of K_m,n and K_n into cubes, Australas. J. Combin. 9, 285–290 (1994).
 ↑ Bryant, D., ElZanati, S. I., Maenhaut, B., and Vanden Eynden, C. Decomposition of complete graphs into 5cubes, J. Combin. Des. 14, 159–166 (2006).
 ↑ ^{4.0} ^{4.1} Buratti, M. (private communication).
 ↑ ^{5.0} ^{5.1} ElZanati, S. I. and Vanden Eynden, C. Decomposing complete graphs into cubes, Discuss. Math. Graph Theory, 26, 141–147 (2006).
 ↑ Kotzig, A. Decompositions of complete graphs into isomorphic cubes, J. Combin. Theory Ser. B, 31, 292–296 (1981).
 ↑ Maheo, M. Strongly graceful graphs, Discrete Math. 29, 39–46 (1980).
 ↑ Wilson, R. M. Decompositions of complete graphs into subgraphs isomorphic to a given graph, Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man. 647–659 (1976).