Weak acyclic coloring and asymmetric coloring games

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12 Scopus citations

Abstract

We introduce the notion of weak acyclic coloring of a graph. This is a relaxation of the usual notion of acyclic coloring which is often sufficient for applications. We then use this concept to analyze the ( a, b )-coloring game. This game is played on a finite graph G, using a set of colors X, by two players Alice and Bob with Alice playing first. On each turn Alice (Bob) chooses a (b) uncolored vertices and properly colors them with colors from X. Alice wins if the players eventually create a proper coloring of G; otherwise Bob wins when one of the players has no legal move. The ( a, b )-game chromatic number of G, denoted ( a, b )- χg ( G ), is the least integer t such that Alice has a winning strategy when the game is played on G using t colors. We show that if the weak acyclic chromatic number of G is at most k then ( 2, 1 )- χg ( G ) {less-than or slanted equal to} frac(1, 2) ( k2 + 3 k ).

Original languageEnglish (US)
Pages (from-to)673-677
Number of pages5
JournalDiscrete Mathematics
Volume306
Issue number7
DOIs
StatePublished - Apr 28 2006

Keywords

  • Acyclic chromatic number
  • Game chromatic number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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