### Abstract

We investigate a competitive version of the coloring number of a graph G = (V, E). For a fixed linear ordering L of V let s (L) be one more than the maximum outdegree of G when G is oriented so that x ← y if x < _{L} y. The coloring number of G is the minimum of s (L) over all such orderings. The (a, b)-marking game is played on a graph G = (V, E) as follows. At the start all vertices are unmarked. The players, Alice and Bob, take turns playing. A play consists of Alice marking a unmarked vertices or Bob marking b unmarked vertices. The game ends when there are no remaining unmarked vertices. Together the players create a linear ordering L of V defined by x < y if x is marked before y. The score of the game is s (L). The (a, b)-game coloring number of G is the minimum score that Alice can obtain regardless of Bob's strategy. The usual (1, 1)-marking game is well studied and there are many interesting results. Our main result is that if G has an orientation with maximum outdegree k then the (k, 1)-game coloring number of G is at most 2k + 2. This extends a fundamental result on the (1, 1)-game coloring number of trees. We also construct examples to show that this bound is tight for many classes of graphs. Finally we prove bounds on the (a, 1)-game coloring number when a < k.

Original language | English (US) |
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Pages (from-to) | 93-107 |

Number of pages | 15 |

Journal | Order |

Volume | 22 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2005 |

### Keywords

- Coloring number
- Competitive coloring
- Planar graph

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

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## Cite this

*Order*,

*22*(2), 93-107. https://doi.org/10.1007/s11083-005-9012-y