Every H -decomposition of K n has a Nearly Resolvable Alternative.

Let H be a fixed graph. An H-decomposition of Kn is a coloring of the edges of Kn 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 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 is χ(L) ≥ n−1 d where d is the average degree of H. We prove that whenever Kn has an Hdecomposition, it also has a decomposition L satisfying χ(L) = n−1 d (1 + on(1)).