The game chromatic number of trees and forests

TL;DR

This paper investigates the game chromatic number of forests, providing criteria for specific cases and examples illustrating the differences between forests with chromatic numbers 2, 3, and 4.

Contribution

It establishes necessary and sufficient conditions for forests with chromatic number 2 and explores the distinctions between forests with chromatic numbers 3 and 4, including key examples.

Findings

Criteria for forests with game chromatic number 2

Minimal example of a forest with chromatic number 4

Forest with maximum degree 3 and chromatic number 4

Abstract

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4.