Relaxed game chromatic number of graphs Original Research Article

This paper discusses a coloring game on graphs. Let k,d be non-negative integers and C a set of k colors. Two persons, Alice and Bob, alternately color the vertices of G with colors from C, with Alice having the first move. A color i is legal for an uncolored vertex x if by coloring x with color i, the subgraph of G induced by those vertices of color i has maximum degree at most d. Each move of Alice or Bob colors an uncolored vertex with a legal color. The game is over if either all vertices are colored, or no more vertices can be colored with a legal color. Aliceʹs goal is to produce a legal coloring which colors all the vertices of G, and Bobʹs goal is to prevent this from happening. We shall prove that if G is a forest, then for k=3,d⩾1, Alice has a winning strategy. If G is an outerplanar graph, then for k=6 and d⩾1, Alice has a winning strategy.