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 [arXiv:1103.5849 ], 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 Gn,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. In this work we show that for the case where F = P` is a long path, the picture is very different. It is not hard to see that m(P`, r) = 1−1/k(P`, r) for an appropriately defined integer k(P`, r), and that the greedy strategy gives a lower bound of k(P`, r) ≥ `r. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in `, and we show that no superpolynomial improvement is possible. ∗An extended abstract of this work will appear in the proceedings of EuroComb ’11. †The author was supported by a fellowship of the Swiss National Science Foundation. the electronic journal of combinatorics 18 (2011), #P163 1