Every H-decomposition of Kn has a Nearly Resolvable Alternative

Noga Alon, Raphael Yuster

Research output: Contribution to journalArticlepeer-review


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)).

Original languageEnglish
Pages (from-to)839-845
Number of pages7
JournalEuropean Journal of Combinatorics
Issue number7
StatePublished - Oct 2000

Bibliographical note

Funding 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


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