We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most $d-1$ edges of the $d$-dimensional hypercube $Q_d$ can be extended to a proper $d$-edge coloring of $Q_d$. Additionally, we characterize which partial edge colorings of $Q_d$ with precisely $d$ precolored edges are extendable to proper $d$-edge colorings of $Q_d$, and consider some related edge precoloring extension problems of hypercubes.

中文翻译：

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超立方体的边缘预着色扩展

我们考虑扩展超立方体的部分边缘着色的问题。特别是，我们通过证明 $d$ 维超立方体 $Q_d$ 的至多 $d-1$ 边的每个适当的部分边着色，获得了著名的埃文斯猜想的正解的类似物，即完成部分拉丁方阵可以扩展到 $Q_d$ 的适当 $d$-edge 着色。此外，我们表征了具有精确 $d$ 预着色边缘的 $Q_d$ 的哪些部分边缘着色可扩展到 $Q_d$ 的正确 $d$-edge 着色，并考虑了超立方体的一些相关边缘预着色扩展问题。