The independence coloring game on graphs
Bo\v{s}tjan Bre\v{s}ar, Da\v{s}a \v{S}tesl

TL;DR
This paper introduces the independence coloring game on graphs, analyzing its properties, variants, and bounds, and demonstrates that the independence game chromatic number can be arbitrarily large for trees.
Contribution
The paper defines four variants of the independence game chromatic number, characterizes graphs with the minimum value, and compares these variants with classical chromatic parameters.
Findings
The invariants lie between the chromatic number and maximum degree plus one.
Graphs with independence game chromatic number equal to 2 are characterized.
The independence game chromatic number of trees can be arbitrarily large.
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 , 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 , where , the players are taking turns in selecting a previously unselected vertex of and giving it color (hence, in each round the selected vertices form an independent set). The game ends when all vertices of 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, , of . In fact, four different versions of the independence game…
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