Symmetric Layer-Rainbow Colorations of Cubes

Abstract

Can we color the n 3 cells of an n × n × n cube L with n 2 colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that L ij = L j i = L ij for distinct i, j, ∈ {1, . . . , n}, and L iij = L jji , L iji = L jij , L ijj = L jii for i, j ∈ {1, . . . , n}? Using transportation networks, we show that such a coloring is possible if and only if n ≡ 0, 2 (mod 3) (with two exceptions, n = 1 and n = 3). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940's. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.

A non-uniform extension of Baranyai's Theorem

“…• Our method might be extendable to the scenario when repetition of subsets are allowed in the hypergraph. In [2], Bahmanian show that the existence of a symmetric Latin cube is equivalent to the existence of a partition into parallel classes of some non-uniform set system. In particular, he determined all n such that…”

Ryser Type Conditions for Extending Colorings of Triples

“…Symmetric latin squares of order |X| can be seen as one-factorizations of `X 1 ˘Y`X 2 ˘, so to account for `X 1 ˘, a further condition is needed (see [11]). Symmetric layer-rainbow latin cubes can be seen as one-factorizations of `X 1 ˘Y 3 `X 2 ˘Y 2 `X 3 ˘ [2]. For example, consider the following one-factorization F of `X 1 ˘Y 3 `X 2 ˘Y 2 `X 3 ˘for X " t1, 2, 3, 4, 5u (We abbreviate sets such as tx, y, zu to xyz).…”