Sharp bounds for decompositions of graphs into complete r-partite subgraphs

If G is a graph on n vertices and r 2 2, we let m,(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, f(G). In determining m,(G), we may assume that no two vertices of G have the same neighbor set. For such reduced graphs G, w e prove that m,(G) 2 log,(n + r l)/r. Furthermore, for each k 2 0 and r 2 2, there is a unique reduced graph G = G(r, k) with m,(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound m,(G) 2 max{n+(G), n-(G)/(r I)}, and show that equality holds if G = G(r, k).