EveryH-decomposition ofKnhas a Nearly Resolvable Alternative

Let H be a fixed graph. An H -decomposition of Knis a coloring of the edges of Knsuch 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 Kn, 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, . . . ,Lt , where any two members belonging to the same subset are vertex-disjoint. A trivial lower bound isχ (L) ≥ n − 1__ dwhere __ d is the average degree of H. We prove that whenever Knhas an H -decomposition, it also has a decomposition L satisfyingχ (L) = n − 1__ d(1 + on(1)).