56,236 research outputs found
On the path-avoidance vertex-coloring game
For any graph and any integer , the \emph{online vertex-Ramsey
density of and }, denoted , is a parameter defined via a
deterministic two-player Ramsey-type game (Painter vs.\ Builder). This
parameter was introduced in a recent paper \cite{mrs11}, 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
). For a large class of graphs , 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
are known. In this work we show that for the case where
is a (long) path, the picture is very different. It is not hard to see that
for an appropriately defined integer
, and that the greedy strategy gives a lower bound of
. 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
- …