Online vertex-coloring games in random graphs

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.