Summary: Every H-decomposition of Kn 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 (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 (L) n-1
where d is the average degree of H. We prove that whenever Kn has an H-
decomposition, it also has a decomposition L satisfying (L) = n-1
(1 + on(1)).
All graphs and hypergraphs considered here are finite, undirected, simple, and have no isolated
vertices. For standard graph-theoretic terminology the reader is referred to . Let H and G be
two graphs. An H-decomposition of G is a coloring of the edges of G, where each color class forms
a copy of H. Each copy is called a member of the decomposition. An H-decomposition of Kn is