### 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 of colors X, with X = r. A color α ∈ X is a legal color for uncolored vertex v if by coloring v with color 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 a partial k-tree, r = k + 1, and d ≥ 4k - 1, then Alice has a winning strategy for this game. In the special case that k = 1, this answers a question of Chou, Wang, and Zhu. We also analyze this strategy for other classes of graphs. In particular, we show that there exists a positive integer d such that Alice can win the game on any planar graph if r = 6.

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

Pages (from-to) | 93-106 |

Number of pages | 14 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 90 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 90, no. 1, pp. 93-106. https://doi.org/10.1016/S0095-8956(03)00077-7

}

TY - JOUR

T1 - A simple competitive graph coloring algorithm II

AU - Dunn, Charles

AU - Kierstead, Henry

PY - 2004/1

Y1 - 2004/1

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 of colors X, with X = r. A color α ∈ X is a legal color for uncolored vertex v if by coloring v with color 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 a partial k-tree, r = k + 1, and d ≥ 4k - 1, then Alice has a winning strategy for this game. In the special case that k = 1, this answers a question of Chou, Wang, and Zhu. We also analyze this strategy for other classes of graphs. In particular, we show that there exists a positive integer d such that Alice can win the game on any planar graph if r = 6.

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 of colors X, with X = r. A color α ∈ X is a legal color for uncolored vertex v if by coloring v with color 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 a partial k-tree, r = k + 1, and d ≥ 4k - 1, then Alice has a winning strategy for this game. In the special case that k = 1, this answers a question of Chou, Wang, and Zhu. We also analyze this strategy for other classes of graphs. In particular, we show that there exists a positive integer d such that Alice can win the game on any planar graph if r = 6.

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UR - http://www.scopus.com/inward/citedby.url?scp=0345868511&partnerID=8YFLogxK

U2 - 10.1016/S0095-8956(03)00077-7

DO - 10.1016/S0095-8956(03)00077-7

M3 - Article

AN - SCOPUS:0345868511

VL - 90

SP - 93

EP - 106

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -