Ore-type graph packing problems

We say that $n$-vertex graphs $G_1,G_2,\ldots,G_k$ pack if there exist injective mappings of their vertex sets onto $[n] = \{1, \ldots,n \}$ such that the images of the edge sets do not intersect. The notion of packing allows one to make some problems on graphs more natural or more general. Clearly, two $n$-vertex graphs $G_1$ and $G_2$ pack if and only if $G_1$ is a subgraph of the complement $\overline{G}_2$ of $G_2$.