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 in polynomial time 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. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.
Highlights
The map-coloring game was conceived by Steven Brams and first published in 1981 by Martin Gardner in Scientific American [11]
Because the necessary and sufficient conditions are trivial for forests with game chromatic number 0 and 1, Question 2 essentially reduces to the existence of a polynomial time algorithm to differentiate between forests with game chromatic number 3 and 4, so the remaining part of the paper investigates these differences
When Bob is playing the k-Expanded Coloring Game (ECG) or k-coloring game, we call coloring a vertex v with α a winning move if v is uncolored, α is legal for v, and coloring v with α will make some vertex uncolorable
Summary
The map-coloring game was conceived by Steven Brams and first published in 1981 by Martin Gardner in Scientific American [11]. The game chromatic number of a graph G, denoted χg(G), is the least t such that Alice has a winning strategy when the t-coloring game is played on G. They proved the stronger result that any forest has game coloring number of at most 4 Their proof implies that if the coloring game is played with an unlimited set of colors, Alice can guarantee each vertex becomes colored before 4 of its neighbors are colored. Because the necessary and sufficient conditions are trivial for forests with game chromatic number 0 and 1, Question 2 essentially reduces to the existence of a polynomial time algorithm to differentiate between forests with game chromatic number 3 and 4, so the remaining part of the paper investigates these differences.
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