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We consider edge decompositions of the n-dimensional hypercube Qn into isomorphic copies of a given graph H. While a number of results are known about decomposing Qn into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if n is even, ℓ < 22 and ℓ divides the number of edges of Qn, then the path of length ℓ decomposes Qn. Tapadia et al. proved that any path of length 2mn, where 2m < n, satisfying these conditions decomposes Qn. Here, we make progress toward resolving Erde’s conjecture by showing that cycles of certain lengths up to 2n+1/n decompose Qn. As a consequence, we show that Qn can be decomposed into copies of any path of length at most 2n/n dividing the number of edges of Qn, thereby settling Erde’s conjecture up to a linear factor
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