The game Grundy number of graphs

Given a graph G=(V,E), two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbours. Alice’s goal is to minimise the total number of colours used in the game, and Bob’s goal is to maximise it. The game Grundy number of G is the number of colours used in the game when both players use optimal strategies. It is proved in this paper that the maximum game Grundy number of forests is 3, and the game Grundy number of any partial 2-tree is at most 7.

[1]  Xuding Zhu,et al.  Lower bounds for the game colouring number of partial k-trees and planar graphs , 2008, Discret. Math..

[2]  Xuding Zhu,et al.  Refined activation strategy for the marking game , 2008, J. Comb. Theory B.

[3]  Hal A. Kierstead,et al.  Competitive Colorings of Oriented Graphs , 2000, Electron. J. Comb..

[4]  Hal A. Kierstead,et al.  Very Asymmetric Marking Games , 2005, Order.

[5]  Hal A. Kierstead,et al.  Orderings on Graphs and Game Coloring Number , 2003, Order.

[6]  Xuding Zhu,et al.  Edge‐partitions of planar graphs and their game coloring numbers , 2002, J. Graph Theory.

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

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

[9]  Jaroslav Nesetril,et al.  On the Oriented Game Chromatic Number , 2001, Electron. J. Comb..

[10]  Pawan Kumar Patel,et al.  Game Coloring Number of Planar Graphs , 2013 .

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

[12]  Xuding Zhu,et al.  The Map-Coloring Game , 2007, Am. Math. Mon..

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

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

[15]  Hal A. Kierstead,et al.  Planar Graph Coloring with an Uncooperative Partner , 1994, Planar Graphs.

[16]  Zsolt Tuza,et al.  Marking Games and the Oriented Game Chromatic Number of Partial k-Trees , 2003, Graphs Comb..

[17]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[18]  Xuding Zhu.  Game coloring the Cartesian product of graphs , 2008 .

[19]  Charles Dunn,et al.  A simple competitive graph coloring algorithm II , 2004, J. Comb. Theory, Ser. B.

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