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 creating a monochromatic copy of some fixed graph F for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors r and a large class of graphs F. This extends earlier work for the case r = 2 by Marciniszyn, Mitsche, and Stojaković. We also prove a similar threshold result for the vertex-coloring analogue of this game. 1

Abstract. 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 on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow. 1.

Abstract. 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

Abstract. Consider the following one-player game. The vertices of a random graph on n vertices are revealed to the player one by one. In each step, also all edges connecting the newly revealed vertex to preceding vertices are revealed. The player has a fixed number of colors at her disposal, and has to assign one of these to each vertex immediately. However, she is not allowed to create any monochromatic copy of some fixed graph F in the process. For n → ∞, we study how the limiting probability that the player can color all n vertices in this online fashion depends on the edge density of the underlying random graph. For a large family of graphs F, including cliques and cycles of arbitrary size, and any fixed number of colors, we establish explicit threshold functions for this edge density. In particular, we show that the order of magnitude of these threshold functions depends on the number of colors, which is in contrast to the corresponding offline coloring problem. 1.

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