Local chromatic number and distinguishing the strength of topological obstructions

Gábor Simonyi, G. Tardos, Sinišia T. Vrećica

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K 2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results of the first two authors.

Original languageEnglish
Pages (from-to)889-908
Number of pages20
JournalTransactions of the American Mathematical Society
Volume361
Issue number2
DOIs
Publication statusPublished - Feb 2009

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Coloring
Chromatic number
Obstruction
Color
Graph in graph theory
Lower bound
Homomorphism
Colouring
Imply
Closed
Line
Vertex of a graph

Keywords

  • Borsuk-ulam theorem
  • Box complex
  • Local chromatic number

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Local chromatic number and distinguishing the strength of topological obstructions. / Simonyi, Gábor; Tardos, G.; Vrećica, Sinišia T.

In: Transactions of the American Mathematical Society, Vol. 361, No. 2, 02.2009, p. 889-908.

Research output: Contribution to journalArticle

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