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

If G is a graph on n vertices and r ≥ 2, we let mr(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, E(G). In determining mr(G), we may assume that no two vertices of G have the same neighbor set. For such reducedgraphs G, we prove that mr(G) ≥ log2 (n + r − 1)/r. Furthermore, for each k ≥ 0 and r ≥ 2, there is a unique reduced graph G = G(r, k) with mr(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound mr(G) ≥ max{n+ (G, n−(G)/(r − 1)}, and show that equality holds if G = G(r, k). © 1996 John Wiley & Sons, Inc.