Edge-colored cube decompositions

Summary.

An edge-colored graph is a graph H together with a function $f:E(H) \mapsto C$ where C is a set of colors. Given an edge-colored graph H, the graph induced by the edges of color c∈C is denoted by H(c). Let G, H, and J be graphs and let μ be a positive integer. A (J, H, G, μ) edge-colored graph decomposition is a set S  =  {H₁,H₂,...,H _t } of edge-colored graphs with color set C  =  {c₁, c₂,..., c _k } such that $H_{i} \cong H$ for 1 ≤  i ≤  t; $H_{i} (c_{j} ) \cong G$ for 1 ≤  i ≤  t and 1 ≤  j ≤  k; and for j = 1, 2,..., k, each edge of J occurs in exactly μ of the graphs H₁(c _j ), H₂(c _j ),..., H _t (c _j ). Let Q₃ denote the 3-dimensional cube. In this paper, we find necessary and sufficient conditions on n, μ and G for the existence of a (K _n ,Q₃,G,μ) edge-colored graph decomposition.