We continue the study of random lifts of graphs initiated in [4]. Here we study the possibility of generating graphs with high edge expansion as random lifts. Along the way, we introduce the method of $\epsilon$-nets into the study of random structures. This enables us to improve (slightly) the known bounds for the edge expansion of regular graphs.

Let $H$ be a fixed graph on $h$ vertices. We say that a graph $G$ is induced$H$-free if it does not contain any induced copy of $H$. Let $G$ be a graph on $n$ vertices and suppose that at least $\epsilon n^2$ edges have to be added to or removed from it in order to make it induced $H$-free. It was shown in [5] that in this case $G$ contains at least $f(\epsilon,h)n^h$ induced copies of $H$, where $1/f(\epsilon,h)$ is an extremely fast growing function in $1/\epsilon$, that is independent of $n$. As a consequence, it follows that for every $H$, testing induced $H$-freeness with one-sided error has query complexity independent of $n$. A natural question, raised by the first author in [1], is to decide for which graphs $H$ the function $1/f(\epsilon,H)$ can be bounded from above by a polynomial in $1/\epsilon$. An equivalent question is: For which graphs $H$ can one design a one-sided error property tester for testing induced $H$-freeness, whose query complexity is polynomial in $1/\epsilon$? We settle this question almost completely by showing that, quite surprisingly, for any graph other than the paths of lengths 1,2 and 3, the cycle of length 4, and their complements, no such property tester exists. We further show that a similar result also applies to the case of directed graphs, thus answering a question raised by the authors in [9]. We finally show that the same results hold even in the case of two-sided error property testers. The proofs combine combinatorial, graph-theoretic and probabilistic arguments with results from additive number theory.

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences $E_N\in\{-1,1\}^N$ in order to measure their ‘level of randomness’. Two of these parameters are the normality measure$\cal{N}(E_N)$ and the correlation measure$C_k(E_N)$of order k, which focus on different combinatorial aspects of $E_N$. In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.

In this paper, we continue the work in this direction and prove a lower bound for the correlation measure $C_k(E_N)$ (k even) for arbitrary sequences $E_N$, establishing one of their conjectures. We also give an algebraic construction for a sequence $E_N$ with small normality measure $\cal{N}(E_N)$.

The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices.

These measures are compared on a class of lattices and are analysed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.

We give results for the age-dependent distribution of vertex degree and number of vertices of given degree in the undirected web-graph process, a discrete random graph process introduced in [8]. For such processes we show that as $k \rightarrow \infty$, the expected proportion of vertices of degree $k$ has power law parameter $1+1/\eta$ where $\eta$ is the limiting ratio of the expected number of edge endpoints inserted by preferential attachment to the expected total degree. The proof for the undirected process generalizes naturally to give similar results for the directed hub-authority process, and an undirected hypergraph process.

We show how to generate labelled and unlabelled outerplanar graphs with $n$ vertices uniformly at random in polynomial time in $n$. To generate labelled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labelled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs.

Next we modify our formulas to also count rooted unlabelled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabelled outerplanar graphs uniformly at random in expected polynomial time.

We define ${\mathcal B}_n$ to be the set of $n$-tuples of the form $(a_0, {\ldots}\,, a_{n-1})$ where $a_j = \pm 1$. If $A \in {\mathcal B}_n$, then we call $A$ a binary sequence and define the autocorrelations of $A$ by $c_k := \sum_{j=0}^{n-k-1} a_j a_{j+k}$ for $0 \leq k \leq n-1$. The problem of finding binary sequences with autocorrelations ‘near zero’ has arisen in communications engineering and is also relevant to conjectures of Littlewood and Erdős on ‘flat’ polynomials with $\pm 1$ coefficients. Following Turyn, we define \[ b(n) := \min_{A \in {\mathcal B}_n} \max_{1 \leq k \leq n-1} |c_k|.\] The purpose of this article is to show that, using some known techniques from discrete probability, we can improve upon the best upper bound on $b(n)$ appearing in the previous literature, and we can obtain both asymptotic and exact expressions for the expected value of $c_k^m$ if the $a_j$ are independent $\pm 1$ random variables with mean 0. We also include some brief heuristic remarks in support of the unproved conjecture that $b(n) = O(\sqrt{n})$.

For a random permutation of $n$ objects, as $n \to \infty$, the process giving the proportion of elements in the longest cycle, the second-longest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected $\ell_1$ distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence.

One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scale-invariant spacing lemma’ for the scale-invariant Poisson processes, proved in this paper.

The theorems of Hindman and van der Waerden belong to the classical theorems of partition Ramsey Theory. The Central Sets Theorem is a strong simultaneous extension of both theorems that applies to general commutative semigroups. We give a common extension of the Central Sets Theorem and Ramsey's theorem.

Let $c$ be a constant and $(e_1,f_1), (e_2,f_2), \dots, (e_{cn},f_{cn})$ be a sequence of ordered pairs of edges on vertex set $[n]$ chosen uniformly and independently at random. Let $A$ be an algorithm for the on-line choice of one edge from each presented pair, and for $i= 1,\hellip,cn$ let $G_A(i)$ be the graph on vertex set $[n]$ consisting of the first $i$ edges chosen by $A$. We prove that all algorithms in a certain class have a critical value $c_A$ for the emergence of a giant component in $G_A(cn) (ie$, if $c \gt c_A$, then with high probability the largest component in $G_A(cn)$ has $o(n)$ vertices, and if $c > c_A$ then with high probability there is a component of size $\Omega(n)$ in $G_A(cn))$. We show that a particular algorithm in this class with high probability produces a giant component before $0.385 n$ steps in the process ($ie$, we exhibit an algorithm that creates a giant component relatively quickly). The fact that another specific algorithm that is in this class has a critical value resolves a conjecture of Spencer.

In addition, we establish a lower bound on the time of emergence of a giant component in any process produced by an on-line algorithm and show that there is a phase transition for the off-line version of the problem of creating a giant component.

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 $n$.

The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$. At time 0 two disjoint finite sets $\xi_1,\xi_2\subset \mathbb{Z}^d$ are infected with type 1 and type 2 infection respectively. An uninfected site then becomes type 1 (2) infected at a rate proportional to the number of type 1 (2) infected nearest neighbours and once infected it remains so forever. The main result in this paper is, loosely speaking, that the choice of the initial sets $\xi_1$ and $\xi_2$ is irrelevant in deciding whether or not the event of mutual unbounded growth for the two infection types has positive probability.

A dominating set $\cal D$ of a graph $G$ is a subset of $V(G)$ such that, for every vertex $v\in V(G)$, either in $v\in {\cal D}$ or there exists a vertex $u \in {\cal D}$ that is adjacent to $v$. We are interested in finding dominating sets of small cardinality. A dominating set $\cal I$ of a graph $G$ is said to be independent if no two vertices of ${\cal I}$ are connected by an edge of $G$. The size of a smallest independent dominating set of a graph $G$ is the independent domination number of $G$. In this paper we present upper bounds on the independent domination number of random regular graphs. This is achieved by analysing the performance of a randomized greedy algorithm on random regular graphs using differential equations.

We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the set of eigenvalues can be recovered under fairly general conditions, e.g., when the graph has a node-transitive automorphism group. The main result is that by observing polynomially many returns, it is possible to estimate the spectral gap of such a graph up to a constant factor.

In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial $P_G(q)$ of a finite graph $G$ of maximal degree $D$ lie in the disk $|q|< K D$, where $K$ is a constant that is strictly smaller than 8.

Let $\Gamma_{k,g}$ be the class of $k$-connected cubic graphs of girth at least $g$. For several choices of $k$ and $g$, we determine a set ${\cal O}_{k,g}$ of graph operations, for which, if $G$ and $H$ are graphs in $\Gamma_{k,g}$, $G\not\cong H$, and $G$ contains $H$ topologically, then some operation in ${\cal O}_{k,g}$ can be applied to $G$ to result in a smaller graph $G'$ in $\Gamma_{k,g}$ such that, on one hand, $G'$ is contained in $G$ topologically, and on the other hand, $G'$ contains $H$ topologically.

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.

Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.

Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

Every node of an undirected connected graph is coloured white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same colour) or 1 (different colours). The aim is to discover a node of the majority colour, or to conclude that there is the same number of black and white nodes. We consider randomized algorithms for this task and establish upper and lower bounds on their expected running time. Our main contribution are lower bounds showing that some simple and natural algorithms for this problem cannot be improved in general.

We consider the problem of reorienting an oriented matroid so that all its cocircuits are ‘as balanced as possible in ratio’. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit $B = \{B^+, B^-\}$ satisfies $B^+, B^- \neq \emptyset$. We show that, for some reorientation, every signed cocircuit satisfies \[1/f(r) \leq |B^+|/|B^-| \leq f(r)\], where $f(r) \leq 14\,r^2\ln(r)$, and $r$ is the rank of the oriented matroid.

In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) $|E|-|V|+1$.

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassen's results about ‘finitary’ duality for infinite graphs to full duality, including his extensions of Whitney's theorem.