Relaxed Game Chromatic Numbers of Complete Multipartite Graphs

Competitive graph coloring is investigated by studying a game with two players, Alice and Bob, on a finite graph G with a set of r colors. Alice and Bob alternately color the vertices of G with legal colors. In the k-relaxed coloring game, a color c is legal for a vertex v if v has at most k neighbors previously colored c. New results about the 0, 1, and 2-relaxed game chromatic numbers will be presented, completely classifying the 0 and 1-relaxed games and partially classifying the 2-relaxed game. These results will be presented in the context of previous research and given a direction of where they need to go next