A simple competitive graph coloring algorithm III

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 X, with |X|=r. A color α∈X is legal for an uncolored vertex v if by coloring v with 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 planar, then Alice has a winning strategy for this game when r=3 and d⩾132. We also show that for sufficiently large d, if G is a planar graph without a 4-cycle or with girth at least 5, then Alice has a winning strategy for the game when r=2.