### 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 )- χ_{g} ( 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 )- χ_{g} ( G ) {less-than or slanted equal to} frac(1, 2) ( k^{2} + 3 k ).

Original language | English (US) |
---|---|

Pages (from-to) | 673-677 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 306 |

Issue number | 7 |

DOIs | |

State | Published - Apr 28 2006 |

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### Keywords

- Acyclic chromatic number
- Game chromatic number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Weak acyclic coloring and asymmetric coloring games.** / Kierstead, Henry.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 306, no. 7, pp. 673-677. https://doi.org/10.1016/j.disc.2004.07.042

}

TY - JOUR

T1 - Weak acyclic coloring and asymmetric coloring games

AU - Kierstead, Henry

PY - 2006/4/28

Y1 - 2006/4/28

N2 - 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 ).

AB - 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 ).

KW - Acyclic chromatic number

KW - Game chromatic number

UR - http://www.scopus.com/inward/record.url?scp=33645885912&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645885912&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2004.07.042

DO - 10.1016/j.disc.2004.07.042

M3 - Article

AN - SCOPUS:33645885912

VL - 306

SP - 673

EP - 677

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 7

ER -