Low diameter monochromatic covers of complete multipartite graphs

Let the diameter cover number, Dᵗᵣ(G), denote the least integer d such that under any r-coloring of the edges of the graph G, there exists a collection of t monochromatic subgraphs of diameter at most d such that every vertex of G is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when G is a complete multipartite graph with at least three parts. We determine exactly the value of D₂²(G) for all complete tripartite graphs G, and almost all complete multipartite graphs with more than three parts.