Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail [9] in 1992. We prove a variety of results relating Rayleigh matroids to other well-known classes – in particular, we show that a binary matroid is Rayleigh if and only if it does not contain $\mathcal{S}_{8}$ as a minor. This has the consequence that a binary matroid is balanced if and only if it is Rayleigh, and provides the first complete proof in print that $\mathcal{S}_{8}$ is the only minor-minimal binary non-balanced matroid, as claimed in [9]. We also give an example of a balanced matroid which is not Rayleigh.

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability $p$, independently of each other, and a deterministic spreading rule with a fixed parameter $k$: if a vacant site has at least $k$ occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on ${\mathbb Z}^d$; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of $p$ for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than $k$, then the critical probability is 1, while it is $1-1/k$ on the $k$-ary tree. A related result is that in any rooted tree $T$ there is a way of erasing $k$ children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree $T'$ has branching number $\mbox{\rm br}(T')\leq \max\{\mbox{\rm br}(T)-k,\,0\}$. We also prove that on any $2k$-regular non-amenable graph, the critical probability for the $k$-rule is strictly positive.

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.

Let $p_c({\mathbb Q}_n)$ and $p_c({\mathbb Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube ${\mathbb Q}_n = \{0,1\}^n$ and on ${\mathbb Z}^n$, respectively. Let $\Omega = n$ for ${\mathbb G} = {\mathbb Q}_n$ and $\Omega = 2n$ for ${\mathbb G} = {\mathbb Z}^n$ denote the degree of ${\mathbb G}$. We use the lace expansion to prove that for both ${\mathbb G} = {\mathbb Q}_n$ and ${\mathbb G} = {\mathbb Z}^n$, \[p_c({\mathbb G}) = \Omega^{-1} + \Omega^{-2} + \frac{7}{2} \Omega^{-3} + O(\Omega^{-4}).\] This extends by two terms the result $p_c({\mathbb Q}_n) = \Omega^{-1} + O(\Omega^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for ${\mathbb Z}^n$.

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 consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let $G_{n,p}$ be a random graph, and let $S$ be a set of $k$ vertices, chosen uniformly at random. Then, let $G_0$ be the graph obtained by deleting all edges connecting two vertices in $S$. Finally, an adversary may add edges to $G_0$ that do not connect two vertices in $S$, thereby producing the instance $G=G_{n,p,k}^*$. We present an algorithm that on input $G=G_{n,p,k}^*$ finds an independent set of size $\geq k$ within polynomial expected time, provided that $k\geq C(n/p)^{1/2}$ for a certain constant $C>0$. Moreover, we prove that in the case $k\leq (1-\varepsilon)\ln(n)/p$ this problem is hard.

We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge is not part of the mystery perfect matching. We will look for the strategy that maximizes the odds of finding the perfect matching without revealing a fixed number of the edges in that perfect matching. For a complete bipartite graph, this is equivalent to finding a mystery permutation via negative guesses with only a fixed number of incorrect negative guesses.

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 $b(n) = O(\sqrt{n})$.

Motivated by a scheduling problem that arises in the study of optical networks, we prove the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler.

Let $k,n$ be two positive integers, let $w_{sj}, 1 \leq s \leq n, 1 \leq j \leq k$ be nonnegative reals satisfying $\sum_{j=1}^k w_{sj}< 1/n$ for every $1 \leq s \leq n$ and let $d_{sj}$ be arbitrary nonnegative reals. Then there are real numbers $x_1, x_2, {\ldots}\,,x_n$ such that for every $j$, $1 \leq j \leq k$, the $n$ cyclic closed intervals $I_s^{(j)}=[x_s+d_{sj},x_s+d_{sj}+w_{sj}]$, $(1 \leq s \leq n)$, where the endpoints are reduced modulo 1, are pairwise disjoint on the unit circle.

The proof is based on some properties of multivariate polynomials and on the validity of the Dyson conjecture.

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.