Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if [sum ]i(i−2)λi>0 then the graph a.s. has a giant component, while if [sum ]i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine ε, λ′0, λ′1 … such that a.s. the giant component, C, has εn+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−[mid ]C[mid ] vertices, and with λ′in′ of them of degree i.

We present a general algebraic technique and discuss some of its numerous applications in combinatorial number theory, in graph theory and in combinatorics. These applications include results in additive number theory and in the study of graph colouring problems. Many of these are known results, to which we present unified proofs, and some results are new.

The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e., the time taken for it to reach its stationary or equilibrium distribution.

The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

The analysis of randomized search heuristics on classes of functions is fundamental to the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of a simple evolutionary algorithm, called (1+1) EA, on the class of linear functions. We improve the previously best known bound in this setting from (1.39+o(1))en ln n to en ln n+O(n) in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutation probabilities p are derived, which imply expected polynomial optimization time as long as p = O((ln n)/n) and p = Ω(n−C) for a constant C > 0, and which are tight if p = c/n for a constant c > 0. As a consequence, the standard mutation probability p = 1/n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Furthermore, the algorithm turns out to be surprisingly robust since the large neighbourhood explored by the mutation operator does not disrupt the search.

A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of the vertices in which every three consecutive vertices form an edge. In this paper we prove an approximate and asymptotic version of an analogue of Dirac's celebrated theorem for graphs: for each γ>0 there exists n0 such that every 3-uniform hypergraph on $n\geq n_0$ vertices, in which each pair of vertices belongs to at least $(1/2+\gamma)n$ edges, contains a Hamiltonian cycle.

We present a practical algorithm for generating random regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at random, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n → ∞. The expected runtime for these ds is [Oscr ](nd2).

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class – an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle [Ann. Probab. 21 1329–1371].

Babai and Sós have asked whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

Consider a random multigraph G* with given vertex degrees d1,. . .,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that where is the integer such that .

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. We prove that any graph G of minimum degree at least three contains a dominating set D of size at most 3|V(G)|/8. A star S is a graph consisting of a centre x and a set of edges from x to S — x. Clearly, a dominating set D for a graph G corresponds to a set of |D| stars which cover V(G). Thus, we show that the vertices of any graph G of minimum degree 3 can be covered by at most 3|V(G)|/8 vertex disjoint stars. We also show that any connected cubic graph G can be covered by [|V(G)|/9] vertex disjoint paths. Both these results are sharp.

We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is the maximum degree of $G$, and the o(1) term tends to zero as $\max\{\sqrt{\Delta}, np\}$ tends to infinity.

Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then

This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).

Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show

Let δ be the minimal degree of G. We show

This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

The main results of this paper are regularity and counting lemmas for 3-uniform hypergraphs. A combination of these two results gives a new proof of a theorem of Frankl and Rödl, of which Szemerédi's theorem for arithmetic progressions of length 4 is a notable consequence. Frankl and Rödl also prove regularity and counting lemmas, but the proofs here, and even the statements, are significantly different. Also included in this paper is a proof of Szemerédi's regularity lemma, some basic facts about quasirandomness for graphs and hypergraphs, and detailed explanations of the motivation for the definitions used.

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ[mid ]I[mid ] for some fixed λ > 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n/n for two given points, that the maximum if one point is fixed and the other varies is 2 log n/n, and that the maximum over all pairs of points is 3 log n/n.

Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.