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 surely (a.a.s.) provided that p ≤ bn−β for some constants b = b(F,k) and β = β(F). This result is essentially best possible because for p ≥ Bn−β, where B = B(F,k) is a large constant, such an edge‐coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n for arbitrary F1,…,Fk.In this article we address the case when F1,…,Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k‐edge‐coloring of Gn,p with p ≤ bn−β for some constant b = b(F1,…,Fk), where β = β(F1,…,Fk) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F1,…,Fk) such that for p ≥ Bn−β the random graph Gn,p a.a.s. does not have a valid k‐edge‐coloring provided the so‐called KŁR‐conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

In this paper we determine the local and global resilience of random graphs $G_{n, p}$ ($p \gg n^{-1}$) with respect to the property of containing a cycle of length at least $(1-\alpha)n$. Roughly speaking, given $\alpha > 0$, we determine the smallest $r_g(G, \alpha)$ with the property that almost surely every subgraph of $G = G_{n, p}$ having more than $r_g(G, \alpha) |E(G)|$ edges contains a cycle of length at least $(1 - \alpha) n$ (global resilience). We also obtain, for $\alpha < 1/2$, the smallest $r_l(G, \alpha)$ such that any $H \subseteq G$ having $\deg_H(v)$ larger than $r_l(G, \alpha) \deg_G(v)$ for all $v \in V(G)$ contains a cycle of length at least $(1 - \alpha) n$ (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of n β (F,r) on the typical duration of this game, where β(F, r) is a function that is strictly increasing in r and satisfies lim r→∞ β(F, r) = 2 − 1/m 2 (F), where n 2−1/m 2 (F) is the threshold of the corresponding offline colouring problem.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. 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.

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.