Online Coloring a Token Graph

We study a combinatorial coloring game between two players, Spoiler and Painter, who alternate turns. First, Spoiler places a new token at a vertex in G, and Painter responds by assigning a color to the new token. Painter must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in G is at most 1) has chromatic number at most w. Painter wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let \(f(w,G)\) be the minimum number of colors needed in an optimal Painter strategy. The game is motivated by a natural online coloring problem on the real line which remains open. A graph G is token-perfect if \(f(w,G) = w\) for each w. We show that a graph is token-perfect if and only if it can be obtained from a bipartite graph by cloning vertices. We also give a forbidden induced subgraph characterization of the class of token-perfect graphs, which may be of independent interest. When G is not token-perfect, determining \(f(w,G)\) seems challenging; we establish \(f(w,G)\) asymptotically for some of the minimal graphs that are not token-perfect.