Simple Minimum (K₄ − e)-coverings of Complete Multipartite Graphs

A decomposition of K_n(g) ∪ Γ, the complete n-partite equipartite graph over gn vertices union a graph Γ (called the excess) that is a subgraph of K_n(g), into edge disjoint copies of a graph G is called a simple minimum group divisible covering of type gⁿ with G if Γ contains as few edges as possible. We examine all possible excesses for simple minimum group divisible (K₄ − e)-coverings. Necessary and sufficient conditions are established for their existence.