Families of Trees Decompose the Random Graph in an Arbitrary Way

Let F ={H1, . . . ,Hk} be a family of graphs. A graph G is called totally F -decomposable iffor every linear combination of the form α1e(H1) + · · · + αke(Hk) = e(G) where each αi is anonnegative integer, there is a coloring of the edges of G with α1 + · · · + αk colors such thatexactly αi color classes induce each a copy of Hi, for i = 1, . . . , k. We prove that if F is anyfixed nontrivial family of trees then logn/n is a sharp threshold function for the property thatthe random graph G(n, p) is totally F -decomposable. In particular, if H is a tree with morethan one edge, then logn/n is a sharp threshold function for the property that G(n, p) contains⌊e(G)/e(H)⌋ edge-disjoint copies of H .