On the path-avoidance vertex-coloring game

For any graph F and any integer r ⩾ 2 , the online vertex-Ramsey density of F and r, denoted m ⁎ ( F , r ) , is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [Mütze, T., T. Rast and R. Spöhel, Coloring random graphs online without creating monochromatic subgraphs, in: Proceedings of the 22nd annual ACM-SIAM Symposium on Discrete Algorithms (SODA ʼ11), 2011, pp. 145–158], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph G n , p ). For a large class of graphs F, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for m ⁎ ( F , r ) are known. s work we show that in the innocent-looking case where F = P ℓ is a (long) path, the picture is very different. As it turns out, for this family of graphs the greedy strategy fails quite badly, and the parameter m ⁎ ( F , r ) exhibits a much more complex behavior than one might expect in view of the other examples mentioned.