Relaxed game chromatic number of graphs

This paper discusses a coloring game on graphs. Let k, d be non-negative integers and C a set of k colors. Two persons, Alice and Bob, alternately color the vertices of G with colors from C, with Alice having the first move. A color i is legal for an uncolored vertex x if by coloring x with color i, the subgraph of G induced by those vertices of color i has maximum degree at most d. Each move of Alice or Bob colors an uncolored vertex with a legal color. The game is over if either all vertices are colored, or no more vertices can be colored with a legal color. Alice's goal is to produce a legal coloring which colors all the vertices of G, and Bob's goal is to prevent this from happening. We shall prove that if G is a forest, then for k = 3, d ≥ 1, Alice has a winning strategy. If G is an outerplanar graph, then for k = 6 and d ≥ 1, Alice has a winning strategy.

[1]  Xuding Zhu.  The Game Coloring Number of Planar Graphs , 1999, J. Comb. Theory, Ser. B.

[2]  Xuding Zhu,et al.  Game chromatic index of k-degenerate graphs , 2001, J. Graph Theory.

[3]  Riste Skrekovski.  List improper colorings of planar graphs with prescribed girth , 2000, Discret. Math..

[4]  Hal A. Kierstead,et al.  Planar Graph Coloring with an Uncooperative Partner , 1991, Planar Graphs.

[5]  U. Faigle,et al.  On the game chromatic number of some classes of graphs , 1991 .

[6]  Hal A. Kierstead,et al.  A simple competitive graph coloring algorithm III , 2004, J. Comb. Theory B.

[7]  L. Cowen,et al.  Defective coloring revisited , 1997 .

[8]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[9]  Xuding Zhu.  The game coloring number of pseudo partial k-trees , 2000, Discret. Math..

[10]  Hans L. Bodlaender.  On the Complexity of Some Coloring Games , 1991, Int. J. Found. Comput. Sci..

[11]  Hal A. Kierstead,et al.  A Simple Competitive Graph Coloring Algorithm , 2000, J. Comb. Theory B.

[12]  Xuding Zhu,et al.  Relaxed Coloring of a Graph , 1998, Graphs Comb..

[13]  Xuding Zhu,et al.  Game chromatic number of outerplanar graphs , 1999, J. Graph Theory.

[14]  Riste Škrekovski.  List Improper Colourings of Planar Graphs , 1999 .

[15]  Douglas R. Woodall,et al.  Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency , 1986, J. Graph Theory.

[16]  M. Frick.  A Survey of (m, k)-Colorings , 1993 .

[17]  Xuding Zhu,et al.  A bound for the game chromatic number of graphs , 1999, Discret. Math..

[18]  É. Sopena,et al.  Acrylic improper colorings of graphs , 1999 .