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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 jX j ¼ r: A color aAX is legal for an uncolored vertex v if by coloring v with a; the subgraph induced by all vertices of color a 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 ¼ 3 and dX132: 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 ¼ 2: r 2004 Elsevier Inc. All rights reserved.

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The (r,d )-relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with , and every neighbor of u that has already been colored with is adjacent to at most d À 1 vertices that have already been colored with . Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)-relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)-relaxed coloring game on G. ß

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