On coloring graphs with locally small chromatic number

H. A. Kierstead, E. Szemerédi, W. T. Trotter

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

In 1973, P. Erdös conjectured that for each kε2, there exists a constant c k so that if G is a graph on n vertices and G has no odd cycle with length less than c k n 1/k, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c k, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.

Original languageEnglish
Pages (from-to)183-185
Number of pages3
JournalCombinatorica
Volume4
Issue number2-3
DOIs
Publication statusPublished - Jun 1984

Fingerprint

Graph Coloring
Coloring
Chromatic number
Odd Cycle
Graph in graph theory
Subgraph
Upper bound

Keywords

  • AMS subject classification (1980): 05C15

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)

Cite this

On coloring graphs with locally small chromatic number. / Kierstead, H. A.; Szemerédi, E.; Trotter, W. T.

In: Combinatorica, Vol. 4, No. 2-3, 06.1984, p. 183-185.

Research output: Contribution to journalArticle

Kierstead, H. A. ; Szemerédi, E. ; Trotter, W. T. / On coloring graphs with locally small chromatic number. In: Combinatorica. 1984 ; Vol. 4, No. 2-3. pp. 183-185.
@article{c28be6b70a1844399d6d06c1bcdc032a,
title = "On coloring graphs with locally small chromatic number",
abstract = "In 1973, P. Erd{\"o}s conjectured that for each kε2, there exists a constant c k so that if G is a graph on n vertices and G has no odd cycle with length less than c k n 1/k, then the chromatic number of G is at most k+1. Constructions due to Lov{\'a}sz and Schriver show that c k, if it exists, must be at least 1. In this paper we settle Erd{\"o}s' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.",
keywords = "AMS subject classification (1980): 05C15",
author = "Kierstead, {H. A.} and E. Szemer{\'e}di and Trotter, {W. T.}",
year = "1984",
month = "6",
doi = "10.1007/BF02579219",
language = "English",
volume = "4",
pages = "183--185",
journal = "Combinatorica",
issn = "0209-9683",
publisher = "Janos Bolyai Mathematical Society",
number = "2-3",

}

TY - JOUR

T1 - On coloring graphs with locally small chromatic number

AU - Kierstead, H. A.

AU - Szemerédi, E.

AU - Trotter, W. T.

PY - 1984/6

Y1 - 1984/6

N2 - In 1973, P. Erdös conjectured that for each kε2, there exists a constant c k so that if G is a graph on n vertices and G has no odd cycle with length less than c k n 1/k, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c k, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.

AB - In 1973, P. Erdös conjectured that for each kε2, there exists a constant c k so that if G is a graph on n vertices and G has no odd cycle with length less than c k n 1/k, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c k, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.

KW - AMS subject classification (1980): 05C15

UR - http://www.scopus.com/inward/record.url?scp=51249185811&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249185811&partnerID=8YFLogxK

U2 - 10.1007/BF02579219

DO - 10.1007/BF02579219

M3 - Article

VL - 4

SP - 183

EP - 185

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2-3

ER -