Every H-decomposition of K_n has a Nearly Resolvable Alternative

Let H be a fixed graph. An H-decomposition of K_n is a coloring of the edges of K_n such that every color class forms a copy of H. Each copy is called a member of the decomposition. The resolution number of an H-decomposition L of K_n, denoted _χ(L), is the minimum number t such that the color classes (i.e., the members) of L can be partitioned into t subsets L₁ , . . . , L_t, where any two members belonging to the same subset are vertex-disjoint. A trivial lower bound is χ(L) ≥ n-1/d where d is the average degree of H. We prove that whenever K_n has an H-decomposition, it also has a decomposition L satisfying χ(L) = n-1/d(1 + 0_n(1)).