The independence coloring game on graphs

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 $G$, 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 $i=1,2,,\ldots$, 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, $\chi_{ig}(G)$, 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.