Abstract

We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph G, and Alice’s goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round i, where i = 1, 2,, … , the players are taking turns in selecting a previously unselected vertex of G and giving it color i (hence, in each round the selected vertices form an independent set). The game ends when all vertices of G are selected (and thus colored), and the total number of rounds during the game when both players are playing optimally with respect to their goals, is called the independence game chromatic number, χ_ig(G), of G. In fact, four different versions of the independence game chromatic number are considered, which depend on who starts a game and who starts next rounds. We prove that the new invariants lie between the chromatic number of a graph and the maximum degree plus 1, and characterize the graphs in which each of the four versions of the game invariant equals 2. We compare the versions of the independence game chromatic number among themselves and with the classical game chromatic number. In addition, we prove that the independence game chromatic number of a tree can be arbitrarily large.

摘要

我们提出了一种新的图上填色游戏，称为独立填色游戏，由两个目标相反的玩家玩。游戏的结果是图G的顶点正确着色，Alice 的目标是在游戏过程中使用尽可能少的颜色，而 Bob 想要最大化颜色的数量。游戏由几轮组成，在第 i 轮中，其中i = 1 , 2 ,, …独立集）。当G的所有顶点都结束时游戏结束被选择（并因此着色），并且当两个玩家都针对他们的目标进行最佳游戏时游戏期间的总轮数称为独立游戏色数，χ _ig ( G )，G. 事实上，考虑了四种不同版本的独立游戏色数，这取决于谁开始游戏以及谁开始下一轮。我们证明了新的不变量位于图的色数和最大度加1之间，并刻画了四个版本的博弈不变量都等于2的图。我们比较了独立博弈色数的版本自己和经典游戏色数。此外，我们证明了树的独立博弈色数可以任意大。