Every H-decomposition of Kn has a Nearly Resolvable Alternative

Noga Alon; Raphael Yuster

doi:10.1006/eujc.2000.0400

Every H-decomposition of K_n has a Nearly Resolvable Alternative

Noga Alon, Raphael Yuster

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let H be a fixed graph. An H-decomposition of K_n is a coloring of the edges of K_n 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 K_n, 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 , . . . , L_t, 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 K_n has an H-decomposition, it also has a decomposition L satisfying χ(L) = n-1/d(1 + 0_n(1)).

Original language	English
Pages (from-to)	839-845
Number of pages	7
Journal	European Journal of Combinatorics
Volume	21
Issue number	7
DOIs	https://doi.org/10.1006/eujc.2000.0400
State	Published - 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

Access to Document

10.1006/eujc.2000.0400

Cite this

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title = "Every H-decomposition of Kn has a Nearly Resolvable Alternative",

abstract = "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)).",

author = "Noga Alon and Raphael Yuster",

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AB - 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)).

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