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 H-decomposition, it also has a decomposition L satisfying χ(L) = n-1/d(1 + 0n(1)).
Bibliographical noteFunding Information:
The authors thank Y. Caro for valuable discussions. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva, Center for Geometry at Tel Aviv University.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics