### Abstract

We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set X, with X =r. A color α∈X is legal for an uncolored vertex v if by coloring v with α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is planar, then Alice has a winning strategy for this game when r=3 and d≥132. We also show that for sufficiently large d, if G is a planar graph without a 4-cycle or with girth at least 5, then Alice has a winning strategy for the game when r=2.

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

Pages (from-to) | 137-150 |

Number of pages | 14 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 92 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2004 |

### Fingerprint

### Keywords

- Planar graph
- Pseudo partial k-tree
- Relaxed game chromatic number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**A simple competitive graph coloring algorithm III.** / Dunn, Charles; Kierstead, Henry.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 92, no. 1, pp. 137-150. https://doi.org/10.1016/j.jctb.2004.03.010

}

TY - JOUR

T1 - A simple competitive graph coloring algorithm III

AU - Dunn, Charles

AU - Kierstead, Henry

PY - 2004/9

Y1 - 2004/9

N2 - We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set X, with X =r. A color α∈X is legal for an uncolored vertex v if by coloring v with α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is planar, then Alice has a winning strategy for this game when r=3 and d≥132. We also show that for sufficiently large d, if G is a planar graph without a 4-cycle or with girth at least 5, then Alice has a winning strategy for the game when r=2.

AB - We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set X, with X =r. A color α∈X is legal for an uncolored vertex v if by coloring v with α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is planar, then Alice has a winning strategy for this game when r=3 and d≥132. We also show that for sufficiently large d, if G is a planar graph without a 4-cycle or with girth at least 5, then Alice has a winning strategy for the game when r=2.

KW - Planar graph

KW - Pseudo partial k-tree

KW - Relaxed game chromatic number

UR - http://www.scopus.com/inward/record.url?scp=4344682453&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344682453&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2004.03.010

DO - 10.1016/j.jctb.2004.03.010

M3 - Article

AN - SCOPUS:4344682453

VL - 92

SP - 137

EP - 150

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -