Relations between the local chromatic number and its directed version

Gábor Simonyi, Gábor Tardos, Ambrus Zsbán

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.

Original languageEnglish
Pages (from-to)318-330
Number of pages13
JournalJournal of Graph Theory
Volume79
Issue number4
DOIs
Publication statusPublished - Aug 1 2015

Keywords

  • fractional colorings
  • local chromatic number
  • oriented graphs

ASJC Scopus subject areas

  • Geometry and Topology

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