A simple competitive graph coloring algorithm II

We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set of colors X, with X = r. A color α ∈ X is a legal color for uncolored vertex v if by coloring v with color α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is a partial k-tree, r = k + 1, and d ≥ 4k - 1, then Alice has a winning strategy for this game. In the special case that k = 1, this answers a question of Chou, Wang, and Zhu. We also analyze this strategy for other classes of graphs. In particular, we show that there exists a positive integer d such that Alice can win the game on any planar graph if r = 6.