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Consider the following one player game on an empty graph with n vertices. The edges are presented one by one to the player in a random order. One of two colors, red or blue, has to be assigned to each edge immediately. The player's object is to color as many edges as possible without creating a monochromatic clique K of some fixed size. We prove a threshold(More)
Consider the following probabilistic one-player game: The board is a graph with n vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all non-edges and is presented to the player, henceforth called Painter. Painter must assign one of r available colors to each edge immediately, where r ≥ 2 is a fixed(More)
Consider the following one-player game on a graph with n vertices. The edges are presented one by one to the player in a random order. One of r available colors has to be assigned to each edge immediately. The player's objective is to color as many edges as possible without creating a monochromatic copy of some fixed graph F. We prove a lower bound of n(More)
Consider the following one-player game on a graph with n vertices. The edges are presented one by one to the player in a random order. One of two colors, red or blue, has to be assigned to each edge immediately. The player's objective is to color as many edges as possible without creating a monochromatic copy of some fixed graph F. We prove an upper bound(More)
The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G 0 as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has(More)
Consider the following one-player game. Starting with the empty graph on n vertices, in every step r new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with r available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid(More)
Consider the following generalized notion of graph colorings: a vertex coloring of graph <i>G</i> is <i>valid w.r.t. some fixed nonempty graph F</i> if no color class induces a copy of <i>F</i> in <i>G</i>, i.e., there is no monochromatic copy of <i>F</i> in <i>G</i>. We propose and analyze an algorithm for computing valid colorings of a random graph(More)
In this paper, we compare the offline versions of three Ramsey-type one-player games that have been studied in an online setting in previous work: the online Ramsey game, the balanced online Ramsey game, and the Achlioptas game. The goal in all games is to color the edges of the random graph ,, according to certain rules without creating a monochromatic(More)
Consider the following problem: For given graphs G and F1,. .. , F k , find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruci´nski studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = · · · = F k = F. They proved that such a coloring exists asymptotically almost(More)