We describe a technique for determining the thresholds for the appearance of cores in random structures. We use it to determine (i) the threshold for the appearance of a k-core in a random r-uniform hypergraph for all r; k * 2; r + k? 4, and (ii) the threshold for the pure literal rule to find a satisfying assignment for a random instance of r-SAT, r * 3.

We give a new derivation of the threshold of appearance of the k-core of a random graph. Our method uses a hybrid model obtained from a simple model of random graphs based on random functions, and the pairing or configuration model for random graphs with given degree sequence. Our approach also gives a simple derivation of properties of the degree sequence of the k-core of a random graph, in particular its relation to multinomial and hence independent Poisson variables. The method is also applied to d-uniform hypergraphs.

We consider random instances of constraint satisfaction problems where each variable has domain size d, and each constraint contains t restrictions on k variables. For each (d; k; t) we determine whether the resolution complexity is a.s. constant, polynomial or exponential in the number of variables. For a particular range of (d; k; t), we determinea sharp threshold for resolution complexity where the resolution complexity drops from a.s. exponential to a.s. poly-nomial when the clause density passes a specific value.

Abstract. We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald [19] on the existence and size of a k-core in G(n, p) and G(n, m), see also Molloy [17] and Cooper [3]. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. 1.

Abstract. In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean λ: = p(n − 1). Motivated by this fact, we introduce the Poisson cloning model GP C(n, p) for random graphs in which the degrees are i.i.d Poisson random variables with mean λ. Then, we first establish a theorem that shows the new model is equivalent to the classical model G(n, p) in an asymptotic sense. Next, we introduce a useful algorithm, called the cut-off line algorithm, to generate the random graph GP C(n, p). The Poisson cloning model GP C(n, p) equipped the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the t-core of G(n, p). This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random hypergraphs and random k-SAT problems. Then, the t-core problem for random hypergraphs and the pure literal algorithm for random k-SAT problems are analyzed. 1

Abstract. We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core. 1.

We prove that for c 2:522 a random graph with n vertices and m = cn edges is not 3-colorable with probability 1 o(1). Similar bounds for non-k-colorability are given for k > 3.

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k-core obeys a law of large numbers as n →∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant k-core. Hence, we deduce corresponding results for the k-core in G(n, p) and G(n, m).

In this paper, we prove that there exists a function ρk = (4 + o(1))k such that G(n, ρ/n) contains a k-regular graph with high probability whenever ρ> ρk. In the case of k = 3, it is also shown that G(n, ρ/n) contains a 3-regular graph with high probability whenever ρ> λ ≈ 5.1494. These are the first constant bounds on the average degree in G(n, p) for the existence of a k-regular subgraph. We also discuss the appearance of 3-regular subgraphs in cores of random graphs. 1

In this thesis, we study the threshold phenomena in the NK landscape, a combinatorial model widely used in the study of genetic algorithms and population genetic dynamics. We establish two random models for the decision problem of the NK landscape model, called the uniform probability model and the fixed ratio model respectively. The aim of the study is to investigate the hardness of the NK landscape model in terms of the theory of threshold phenomena and phase transitions. We show theoretically that the uniform probability model is trivially insoluble as the problem size tends to infinity. For the fixed ratio model, we establish two upper bounds of insolubility on the control parameter of the model above which the problems are asymptotically insoluble with probability 1. We show that instances with parameters above the upper bounds contain some easy subproblems such as 2-SAT, and hence can be solved by polynomial algorithms. The fixed ratio model is also studied empirically. The experimental results show that there is a threshold phenomenon in the model and our upper bound on the threshold is tight. From the experiments, we also observe that random instances of the xed ratio model are also typically easy in the soluble region and phase transition region.