Abstract:We establish a three-dimensional analogue of the classical theorem that a Latin square of order \(m\) can be embedded in a Latin square of order \(n\) if and only if \(n \ge 2m\). Let \(L\) be an \(n\times n\times n\) array. A {\it layer} of \(L\) is obtained by fixing one coordinate. If \(L\) is filled with \(n^2\) symbols so that every layer contains each symbol exactly once, then \(L\) is called a {\it layer-rainbow cube}. If \(L\) is filled with \(n\) symbols and every layer is a Latin square, then \(L\) is called a {\it layer-Latin cube}. Relatively little is known about embedding partial layer-Latin cubes, and the existing results are far from optimal with respect to the order of the containing cube. In contrast, no embedding results appear to be known for layer-rainbow cubes. We resolve this problem completely by proving that a layer-rainbow cube of order \(m\) can be embedded in a layer-rainbow cube of order \(n\) if and only if \(n \ge 2m\). Equivalently, our result may be viewed as an embedding theorem for one-factorizations of complete tripartite \(3\)-uniform hypergraphs.
From: Amin Bahmanian [view email]
[v1]
Wed, 14 Sep 2022 04:13:59 UTC (8 KB)
[v2]
Sun, 16 Aug 2026 05:44:35 UTC (401 KB)
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