Let G = (G, +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets A − B, while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set A ⊂ G is replaced by a discrete random variable X taking values in G, and the cardinality |A| is replaced by the Shannon entropy H(X). In particular, we classify those random variables X which have small doubling in the sense that H(X1 + X2) = H(X) + O(1) when X1, X2 are independent copies of X, by showing that they factorize as X = U + Z, where U is uniformly distributed on a coset progression of bounded rank, and H(Z) = O(1).

When G is torsion-free, we also establish the sharp lower bound , where o(1) goes to zero as H(X) → ∞.

We give results on the strong connectivity for spaces of sparse random digraphs specified by degree sequence. A full characterization is provided, in probability, of the fan-in and fan-out of all vertices including the number of vertices with small ($o(n)$) and large ($cn$) fan-in or fan-out. We also give the size of the giant strongly connected component, if any, and the structure of the bow-tie digraph induced by the vertices with large fan-in or fan-out. Our results follow a direct analogy of the extinction probabilities of classical branching processes.

Let k be a positive integer and let G be a graph. Suppose a list S(v) of positive integers is assigned to each vertex v, such that

(1) [mid ]S(v)[mid ] = 2k for each vertex v of G, and

(2) for each vertex v, and each c ∈ S(v), the number of neighbours w of v for which c ∈ S(w) is at most k.

Then we prove that there exists a proper vertex colouring f of G such that f(v) ∈ S(v) for each v ∈ V(G). This proves a weak version of a conjecture of Reed.

Let G be a graph with maximum degree Δ(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies

where o(l) goes to zero as Δ(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number:

provided g(G) < 4.

We analyse Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on ${\mathbb Z}^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,{\bf x}) : {\bf x} \in {\mathbb Z}^d\}$, where $p(n,{\bf x})$ is the probability that a walk beginning from the origin arrives at ${\bf x}$ at time $n$.

Let G be a graph and P(G, t) be the chromatic polynomial of G. It is known that P(G, t) has no zeros in the intervals (−∞, 0) and (0, 1). We shall show that P(G, t) has no zeros in (1, 32/27]. In addition, we shall construct graphs whose chromatic polynomials have zeros arbitrarily close to 32/27.

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

We define a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sufficient conditions on the edge weights for this expansion not to depend on the order used. We give a contraction-deletion formula for W analogous to that for the Tutte polynomial, and show that any coloured graph invariant satisfying such a formula can be obtained from W. In particular, we show that generalizations of the Tutte polynomial obtained from its rank generating function formulation, or from a random cluster model, can be obtained from W. Finally, we find the most general conditions under which W gives rise to a link invariant, and give as examples the one-variable Jones polynomial, and an invariant taking values in ℤ/22ℤ.

We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let $G$ be any connected, undirected graph with $n$ nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph $G$: in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round.

We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for $G$, the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regard to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only $\O(n \log (\diam(G)))$ against restricted as well as unrestricted rabbits. This bound is close to optimal since $\Omega(n)$ is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits.

For a graph H and an integer r ≥ 2, the induced r-size-Ramsey number of H is defined to be the smallest integer m for which there exists a graph G with m edges with the following property: however one colours the edges of G with r colours, there always exists a monochromatic induced subgraph H′ of G that is isomorphic to H. This is a concept closely related to the classical r-size-Ramsey number of Erdős, Faudree, Rousseau and Schelp, and to the r-induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rödl, and Trotter. Here, we prove a result that implies that the induced r-size-Ramsey number of the cycle Cℓ is at most crℓ for some constant cr that depends only upon r. Thus we settle a conjecture of Graham and Rödl, which states that the above holds for the path Pℓ of order ℓ and also generalise in part a result of Bollobás, Burr and Reimer that implies that the r-size Ramsey number of the cycle Cℓ is linear in ℓ Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemerédi.

Szemerédi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

Suppose that a process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always bounded above by d. We prove that if n → ∞ with d fixed, then with probability tending to 1, the final result of this process is a graph with ⌊nd / 2⌋ edges.

A question recently posed by Häggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

We say that α ∈ [0, 1) is a jump for an integer r ≥ 2 if there exists c(α) > 0 such that for all ϵ > 0 and all t ≥ 1, any r-graph with n ≥ n0(α, ϵ, t) vertices and density at least α + ϵ contains a subgraph on t vertices of density at least α + c.

The Erdős–Stone–Simonovits theorem [4, 5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r ≥ 3, every α ∈ [0, r!/rr) is a jump. Moreover he made his famous ‘jumping constant conjecture’, that for all r ≥ 3, every α ∈ [0, 1) is a jump. Frankl and Rödl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r ≥ 3.

We use Razborov's flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r ≥ 3 in the interval [r!/rr, 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump.

We also give an improved upper bound for the Turán density of K4− = {123, 124, 134}: π(K4−) ≤ 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.

The maximal zero-free intervals for chromatic polynomials of graphs are precisely (−∞, 0), (0, 1), (1, 32/27]. We also investigate the distribution of zeros of chromatic polynomials in various classes of graphs closed under minors. For example, the zeros of chromatic polynomials of graphs of tree-width at most k consist of 0, 1 and a dense subset of the interval (32/27, k].

We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures (of Lachlan and Cherlin) to the classification of Ramsey classes. As particular instances of this approach we characterize all Ramsey classes of graphs, tournaments and partial ordered sets. We fully characterize all monotone Ramsey classes of relational systems (of any type). We also carefully discuss the role of (admissible) orderings which lead to a new classification of Ramsey properties by means of classes of order-invariant objects.

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).

Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$, where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdös and Nosal respectively, we prove that every non-bipartite graph of order and size contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$, and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$, where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

A Fano configuration is the hypergraph of 7 vertices and 7 triplets defined by the points and lines of the finite projective plane of order 2. Proving a conjecture of T. Sós, the largest triple system on $n$ vertices containing no Fano configuration is determined (for $n> n_1$). It is 2-chromatic with $\binom{n}{3}-\binom{\lfloor n/2 \rfloor}{3} -\binom{\lceil n/2 \rceil}{3}$ triples. This is one of the very few nontrivial exact results for hypergraph extremal problems.

We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graph G are assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring of G with at most k colours (for some given k)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.