The Map-Coloring Game

1. INTRODUCTION. Suppose that Alice wants to color a planar map using four colors in a proper way, that is, so that any two adjacent regions get different colors. Despite the fact that she knows for certain that it is eventually possible, she may fail in her first attempts. Indeed, there are usually many proper partial colorings not extend-able to proper colorings of the whole map. Thus, if she is unlucky, she may accidentally create such a bad partial coloring. Now suppose that Alice asks Bob to help her in this task. They color the regions of a map alternately, with Alice going first. Bob agrees to cooperate by respecting the rule of a proper coloring. However, for some reason he does not want the job to be completed—his secret aim is to achieve a bad partial coloring. (For instance, he may wish to start the coloring procedure over and over again just to stay longer in Alice's company.) Is it possible for Alice to complete the coloring somehow, in spite of Bob's insidious plan? If not, then how many additional colors are needed to guarantee that the map can be successfully colored, no matter how clever Bob is? This map-coloring game was invented about twenty-five years ago by Steven J. Brams with the hope of finding a game-theoretic proof of the Four Color Theorem, avoiding perhaps the use of computers. Though this approach has not been successful, at least we are left with a new, intriguing map-coloring problem: What is the fewest number of colors allowing a guaranteed win for Alice in the map-coloring game in the plane? Brams's game was published by Martin Gardner in his " Mathematical Games " column in Scientific American in 1981. Surprisingly, it remained unnoticed by the graph-theoretic community until ten years later, when it was reinvented by Hans L. Bodlaen-der [1] in the wider context of general graphs. In this version Alice and Bob play as before by coloring properly the vertices of a graph G. The game chromatic number χ g (G) of G is the smallest number of colors for which Alice has a winning strategy. As every map is representable by a graph whose edges correspond to adjacent regions of the map, Brams's question is equivalent to determining the game chromatic number of planar graphs. Since then the problem has been analyzed in serious combinatorial journals and gained the …