Minimum H-decompositions of graphs: Edge-critical case

For a given graph H let ϕH(n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that ϕH(n)=ex(n,H) for χ(H)⩾3 and all sufficiently large n, where ex(n,H) denotes the maximum size of a graph on n vertices not containing H as a subgraph. In this article, their conjecture is verified for all edge-critical graphs. Furthermore, it is shown that the graphs maximizing ϕH(n) are (χ(H)−1)-partite Turán graphs.