Chromatic numbers of graphs — large gaps

Assaf Rinot

doi:https://doi.org/10.1007/s00493-014-3074-7

Chromatic numbers of graphs — large gaps

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

Abstract

We say that a graph G is (ℵ₀,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ₀ for any subgraph G′ of G of size < |G|. The main result of this paper reads as follows. If □_λ+CH_λ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ₀,κ)-chromatic graph of size λ⁺. We also study (ℵ₀,λ⁺)-chromatic graphs of size λ⁺. In particular, it is proved that if 0^# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ₀,λ⁺)-chromatic graph of size λ⁺.

Original language	English
Pages (from-to)	215-233
Number of pages	19
Journal	Combinatorica
Volume	35
Issue number	2
DOIs	https://doi.org/10.1007/s00493-014-3074-7
State	Published - 16 Apr 2015
Externally published	Yes

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Computational Mathematics

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title = "Chromatic numbers of graphs — large gaps",

abstract = "We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|. The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+. We also study (ℵ0,λ+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0,λ+)-chromatic graph of size λ+.",

author = "Assaf Rinot",

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doi = "https://doi.org/10.1007/s00493-014-3074-7",

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AU - Rinot, Assaf

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N2 - We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|. The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+. We also study (ℵ0,λ+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0,λ+)-chromatic graph of size λ+.

AB - We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|. The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+. We also study (ℵ0,λ+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0,λ+)-chromatic graph of size λ+.

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VL - 35

SP - 215

EP - 233

JO - Combinatorica

JF - Combinatorica

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ER -

Chromatic numbers of graphs — large gaps

Abstract

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