We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ 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, $\ell <2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al. proved that any path of length $2^{m}n$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde’s conjecture by showing that cycles of certain lengths up to $2^{n+1}∕n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^n∕n$ dividing the number of edges of $Q_n$, thereby settling Erde’s conjecture up to a linear factor.

我们考虑了边的分解 $ñ$维超立方体 ${问}_{ñ}$ 变成给定图的同构副本 $H$。虽然有许多关于分解的结果${问}_{ñ}$在各种类别的图中，给定长度的路径和循环的最简单情况还远远没有被理解。埃尔德（Erde）的一个猜想断言，如果$ñ$ 甚至， $\ell <{\mathrm{2个}}^{ñ}$ 和 $\ell$ 划分的边数 ${问}_{ñ}$，然后是长度的路径 $\ell$ 分解 ${问}_{ñ}$。Tapadia等。证明任何长度的路径${\mathrm{2个}}^{米}ñ$， 在哪里 ${\mathrm{2个}}^{米}<ñ$，满足这些条件分解 ${问}_{ñ}$。在这里，我们通过显示一定长度的周期直到${\mathrm{2个}}^{ñ+\mathrm{1个}}∕ñ$ 分解 ${问}_{ñ}$。结果，我们证明了${问}_{ñ}$ 最多可以分解为任何长度的路径的副本 ${\mathrm{2个}}^{ñ}∕ñ$ 除以的边数 ${问}_{ñ}$，从而使Erde猜想的线性度最高。