The Relaxed Game Chromatic Number Of Outerplanar Graphs

The concept of the relaxed game chromatic number of graphs was introduced by Chou, Wang and Zhu in [21]. This is an interesting definition in graph theory. It can connect coloring problem with game theory. Plenty of research and exploration has been done on this field in the last ten years. Many achievements have been gained. Meanwhile, a considerable number of unsolved problems have been remained. In this paper first we systematically summarize the results of the research of the relaxed game chromatic number of graphs. Then by giving Alice a strategy which is made up of jumping steps and coloring rules, we fully detail that if suppose G = (V, E) is an outerplanar graph, and 1 < d < 4, where d is an integer and k + d = 7, then for the (k, d)-relaxed coloring game on G, Alice will win, namely, Xg(G) < k. The result improves the preceding result from identification [22], Finally, we discuss two cases, namely, (6,1)- relaxed coloring game and (2, 5)- relaxed coloring game on G.