Vector clique decompositions

Let F_k be the set of all graphs on k vertices. For a graph G, a k-decomposition is a set of induced subgraphs of G, each isomorphic to an element of F_k, such that each pair of vertices of G is in exactly one element of the set. It is a fundamental result of Wilson that for all n = |V (G)| sufficiently large, G has a k-decomposition if and only if G is k-divisible, namely k−1 divides n−1 and^k₂ dividesⁿ₂ . Let v ∈ R^|Fk^| be indexed by F_k. For a kdecomposition L of G, let ν_v(L) =^P_F∈Fk v_Fd_L,F where d_L,F is the fraction of elements of L that are isomorphic to F. Let ν_v(G) = max_L ν_v(L) and ν_v(n) = min{ν_v(G): |V (G)| = n} The sequence ν_v(n) has a limit so let ν_v = lim_n→∞ ν_v(n). Replacing kdecompositions with their fractional relaxations, one obtains the (polynomial time computable) fractional analogue ν_v^∗(G) and the corresponding fractional values ν_v^∗(n) and ν_v^∗. Our first main result is that for each v ∈ R^|Fk| ν_v = ν_v^∗ . Furthermore, there is a polynomial time algorithm that produces a decomposition L of a k-decomposable graph such that ν_v(L) ≥ ν_v − o_n(1). A similar result holds when F_k is the family of all tournaments on k vertices or when F_k is the family of all edge-colorings of K_k. We use these results to obtain new and improved bounds on several decomposition results. For example, we prove that every n-vertex tournament which is 3-divisible (namely n = 1, 3 mod 6) has a triangle decomposition in which the number of directed triangles is less than 0.0222n²(1 + o(1)) and that every 5-decomposable n-vertex graph has a 5-decomposition in which the fraction of cycles of length 5 is o_n(1).