  In this paper, two open conjectures are disproved. One conjecture regards
independent coverings of sparse partite graphs, whereas the other conjecture
regards orthogonal colourings of tree graphs. A relation between independent
coverings and orthogonal colourings is established. This relation is applied to
find independent coverings of some sparse partite graphs. Additionally, a
degree condition providing the existence of an independent covering in the case
where the graph has a square number of vertices is found.
