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

The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erdős-Rényi process (ER). It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f (c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of ER are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer [1] gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke [10] very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of ER-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.

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 to be selected, and the remaining r − 1 edges are discarded. Thus after N steps, we have seen rN edges, and selected exactly N out of these to create a graph G N . In a recent paper by Krivelevich, Loh, and Sudakov (2009) [11], the problem of avoiding a copy of some fixed graph F in G N for as long as possible is considered, and a threshold result is derived for some special cases. Moreover, the authors conjecture a general threshold formula for arbitrary graphs F . In this work we disprove this conjecture and give the complete solution of the problem by deriving explicit threshold functions N 0 (F , r, n) for arbitrary graphs F and any fixed integer r. That is, we propose an edge selection strategy that a.a.s. (asymptotically almost surely, i.e. with probability 1 − o(1) as n → ∞) avoids creating a copy of F for as long as N = o(N 0 ), and prove that any online strategy will a.a.s. create such a copy once N = ω(N 0 ).

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n 1−ε , ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(n log log n) guesses. This bound is valid also when only black answer-pegs are used. It improves the O(n log n) bound first proven by Chvátal (Combinatorica 3 (1983), [325][326][327][328][329]. We also show that if both black and white answer-pegs are used, then the O(n log log n) bound holds for up to n 2 log log n colors. These bounds are almost tight as the known lower bound of Ω(n) shows. Unlike for k ≤ n 1−ε , simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal non-adaptive strategy (deterministic or randomized) needs Θ(n log n) guesses.

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.