Reto Spöhel

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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 on(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)
We study the following two problems: i) Given a random graph Gn,m (a graph drawn uniformly at random from all graphs on n vertices with exactly m edges), can we color its edges with r colors such that no color class contains a component of size Θ(n)? ii) Given a random graph Gn,m with a random partition of its edge set into sets of size r, can we color its(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 G0 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 to(More)
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(More)
In this paper, we compare the offline versions of three Ramsey-type oneplayer 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 Gn,m according to certain rules without creating a monochromatic(More)
Consider the following problem: For given graphs G and F1, . . . , Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruciński studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = · · · = Fk = F . They proved that such a coloring exists asymptotically almost(More)