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### The Game Coloring Number of Planar Graphs

Xuding Zhu
1999 Journal of combinatorial theory. Series B (Print)
We show that the game coloring number of a planar graph is at most 19.  ...  This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph.  ...  Since the acyclic chromatic number of a planar graph is at most 5, it follows that the game chromatic number of a planar graph is at most 30.  ...

### Bounds for the game coloring number of planar graphs with a specific girth [article]

Keaitsuda Maneeruk Nakprasit, Kittikorn Nakprasit
2016 arXiv   pre-print
Let col_g(G) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs H as col_g(H) := { col_g(G):G ∈H}.  ...  Let P_k be the family of planar graphs of girth at least k. We show that col_g(P_7) ≤ 5. This result extends a result about the coloring number by Wang and Zhang WZ11 ( col_g(P_8) ≤ 5).  ...  The second author was supported by the Commission on Higher Education and the Thailand Research Fund under grant RSA5780014.  ...

### The Hats game. The power of constructors [article]

Aleksei Latyshev, Konstantin Kokhas
2021 arXiv   pre-print
We present an example of a planar graph for which the sages win for k = 14. We also give an easy proof of the theorem about the Hats game on "windmill" graphs.  ...  We analyze the following general version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of k colors.  ...  The relation between the number HG(G) and the planarity of graph G is one of the open problems. Namely, does there exist a planar graph with an arbitrary large hat guessing number?  ...

### The Map-Coloring Game

Tomasz Bartnicki, Jarosław Grytczuk, H. A. Kierstead, Xuding Zhu
2007 The American mathematical monthly
We would like to thank Steven Brams for useful materials and comments on the origin of the map-coloring game.  ...  The research of the fourth author is supported in part by the National Science Council under grant NSC95-2115-M-110-013-MY3.  ...  2-coloring numbers. proved the Burr-Erdős conjecture for planar graphs by showing that the 2-coloring number of any planar graph is at most 761.  ...

### Planar graph coloring with an uncooperative partner

H. A. Kierstead, W. T. Trotter
1994 Journal of Graph Theory
We show that the game chromatic number of a planar graph is at most 33.  ...  In particular, the game chromatic number of a graph is bounded in terms of its genus.  ...  ACKNOWLEDGMENT The authors would like to express their appreciation to Nate Dean, Paul Seymour, Neil Robertson, and Robin Thomas for stimulating conversations on the interplay between game chromatic number  ...

### A bound for the game chromatic number of graphs

Thomas Dinski, Xuding Zhu
1999 Discrete Mathematics
the game chromatic number of planar graphs.  ...  In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the game chromatic number of a planar graph is at most 30, which improves the previous known upper bound for  ...  This implies, in particular, that the game chromatic number of a planar graph is at most 30.  ...

### Planar graphs decomposable into a forest and a matching

Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh
2009 Discrete Mathematics
He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9.  ...  We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching.  ...  The third author's research was supported in part by the grant DMS-0650784.  ...

### The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses [article]

Clément Charpentier
2015 arXiv   pre-print
We denote by χ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P.  ...  Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5.  ...  that every (1, k)-decomposable graph has χ g (G) ≤ k + 4, then deduced upper bounds for the game chromatic number of planar graphs with given girth.  ...

### Edge-disjoint odd cycles in graphs with small chromatic number

Claude Berge, Bruce Reed
1999 Annales de l'Institut Fourier
For 4-colorable graphs, and in particular for planar graphs, no simple structural property has been found so far to see that the game is unfair. The Konig property.  ...  At the end of the game, all the edges of G have been colored; the red edges define a partial graph GR with no odd cycles and the blue edges define a partial graph GB with no odd cycles.  ...

### On-line Ramsey Theory

J. A. Grytczuk, M. Hałuszczak, H. A. Kierstead
2004 Electronic Journal of Combinatorics
The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game.  ...  In particular, we prove that Builder has a winning strategy for any \$k\$-colorable graph \$H\$ in the game played on \$k\$-colorable graphs.  ...  Let n be the number of vertices of a 3-colorable graph F on which Builder wins against Painter armed in c − 1 colors.  ...

### Page 3162 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews
The question whether planar graphs have bounded game chro- matic number is resolved in this paper: it is shown that the game chromatic number of a planar graph is at most 33.  ...  In particular, the game chro- matic number of a game is bounded in terms of its genus.  ...

### The game coloring number of pseudo partial k-trees

Xuding Zhu
2000 Discrete Mathematics
We discuss the game coloring number (as well as the game chromatic number) of (a; b)-pseudo partial k-trees, and prove that the game coloring number of an (a; b)-pseudo partial k-tree is at most 3k + 2a  ...  , outerplanar graphs and planar graphs.  ...  It follows from Corollary 2 that the game coloring number of an outer-planar graph is at most 8. It was proved in  that the game coloring number of an outer-planar graph is at most 7.  ...

### The game of arboricity

Tomasz Bartnicki, Jaroslaw Grytczuk, Hal Kierstead
2005 Discrete Mathematics & Theoretical Computer Science
International audience Using a fixed set of colors \$C\$, Ann and Ben color the edges of a graph \$G\$ so that no monochromatic cycle may appear.  ...  The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.  ...  The game of arboricity Introduction We consider the following graph coloring game. Ann and Ben alternately color the edges of a graph G using a fixed set of colors C.  ...

### The coloring game on planar graphs with large girth, by a result on sparse cactuses

Clément Charpentier
2017 Discrete Mathematics
We denote by χ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P.  ...  Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5.  ...  This ends the proof of Theorem 1.  ...

### Page 8410 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews
This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chro- matic number of such graphs.  ...  We discuss the game coloring number (as well as the game chromatic number) of (a,b)-pseudo partial k-trees, and prove that the game color- ing number of an (a, b)-pseudo partial k-tree is at most 3k +  ...
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