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We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Q_d, and lists of allowed colors for the non-colored edges of Q_d, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.

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.

We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant y>0 such that if n=2^k and A is a 3-dimensional n×n×n array where every cell contains at most γn symbols, and every symbol occurs at most γn times in every line of A, then A is avoidable; that is, there is a Latin cube L of order n such that for every 1 ≤ i,j,k ≤ n, the symbol in position (i,j,k) of L does not appear in the corresponding cell of A.