Proof of the Alon-Yuster conjecture

János Komlós; Gábor N. Sárközy; Endre Szemerédi

doi:10.1016/S0012-365X(00)00279-X

Proof of the Alon-Yuster conjecture

János Komlós, Gábor N. Sárközy, Endre Szemerédi

School of Arts and Sciences, Mathematics

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

79 Scopus citations

Abstract

In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n₀(H) such that if n ≥ n₀(H) and G is a graph with hn vertices and minimum degree at least (1 - 1/k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h₁ ≤ h₂ ≤ ⋯ ≤ h_k, then the conjecture is true with c(H)=h_k+h_k-1 - 1.

Original language	English (US)
Pages (from-to)	255-269
Number of pages	15
Journal	Discrete Mathematics
Volume	235
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(00)00279-X
State	Published - May 28 2001

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/S0012-365X(00)00279-X

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abstract = "In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n ≥ n0(H) and G is a graph with hn vertices and minimum degree at least (1 - 1/k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h1 ≤ h2 ≤ ⋯ ≤ hk, then the conjecture is true with c(H)=hk+hk-1 - 1.",

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AU - Komlós, János

AU - Sárközy, Gábor N.

AU - Szemerédi, Endre

PY - 2001/5/28

Y1 - 2001/5/28

N2 - In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n ≥ n0(H) and G is a graph with hn vertices and minimum degree at least (1 - 1/k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h1 ≤ h2 ≤ ⋯ ≤ hk, then the conjecture is true with c(H)=hk+hk-1 - 1.

AB - In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n ≥ n0(H) and G is a graph with hn vertices and minimum degree at least (1 - 1/k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h1 ≤ h2 ≤ ⋯ ≤ hk, then the conjecture is true with c(H)=hk+hk-1 - 1.

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Proof of the Alon-Yuster conjecture

Abstract

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