We call a tree parameter additive if it can be determined recursively as the sum of the parameter values of all branches, plus a certain toll function. In this paper, we prove central limit theorems for very general toll functions, provided that they are bounded and small on average. Simply generated families of trees are considered as well as Pólya trees, recursive trees and binary search trees, and the results are illustrated by several examples of parameters for which we prove normal or log-normal limit laws.

Given complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

The theta graph ${\Theta _{\ell ,t}}$ consists of two vertices joined by t vertex-disjoint paths, each of length $\ell $ . For fixed odd $\ell $ and large t, we show that the largest graph not containing ${\Theta _{\ell ,t}}$ has at most ${c_\ell }{t^{1 - 1/\ell }}{n^{1 + 1/\ell }}$ edges and that this is tight apart from the value of ${c_\ell }$ .

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and is ‘attacked by an adversary’. At time t, we add a new vertex xt and m random edges incident with xt, where m is constant. The neighbours of xt are chosen with probability proportional to degree. After adding the edges, the adversary is allowed to delete vertices. The only constraint on the adversarial deletions is that the total number of vertices deleted by time n must be no larger than δn, where δ is a constant. We show that if δ is sufficiently small and m is sufficiently large then with high probability at time n the generated graph has a component of size at least n/30.

We investigate the asymptotic behaviour of the average displacement of the simple random walk on the Sierpiński graph. The existence of an oscillating factor in this asymptotics is shown rigorously. The proof depends mainly on the analysis of the corresponding generating function. Using a functional equation and techniques from complex analysis we obtain the desired properties of this generating function.

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

Let $G$ be a cyclic group of order $n$ and let $\mu = \{x_1,x_2, \dots, x_m\}$ be a sequence of elements of $G$. Let $k$ be the number of distinct values taken by the sequence $\mu$. Let $n\wedge \mu$ be the set of the $n$-subsequence sums.

We show that one of the following conditions holds:

1. $\mu$ has a value repeated $n-k+3$ times

2. $n\wedge \mu$ contains a non-null subgroup

3. $|n\wedge \mu|\geq m-n+k-2.$

We conjecture that the last condition could be improved to $|n\wedge \mu|\geq m-n+k-1$. This conjecture generalizes several known results. We also obtain a generalization of a recent result due to Bollobás and Leader.

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph have no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is essentially the same for all values of k that satisfy both k → ∞ and k =o(n).

Let G be a planar graph without 6-cycles. We investigate structural properties of G and show that G is edge-(Δ(G) + 1)-choosable when its maximum degree Δ(G) is not 5. We also study the 3-degeneracy property of G.

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede–Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lück's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.

We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

Let S be a set of m clauses each containing three literals chosen at random in a set {p1, ¬p1,…,pn, ¬pn} of n propositional variables and their negations. Let be the set of all such S with m = cn for a fixed c > 0. We show, improving significantly over the first moment upper bound , that if m and n tend to infinity with , then almost all are unsatisfiable.

In the group testing problem the aim is to identify a small set of k ⁓ nθ infected individuals out of a population size n, 0 < θ < 1. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result if and only if at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition minf for non-adaptive group testing, where all tests are conducted in parallel. Thus with more than minf tests the infected individuals can be identified in polynomial time with high probability, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.

This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $\nu$, every fixed sample graph $L_\nu$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In [9] we proved that, if the frequency of just one fixed $L_\nu$ – as a not necessarily induced subgraph – in every ‘large’ induced subgraph $F_h\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here we shall investigate the analogous problem for induced subgraphs $L_\nu$. In such cases $(G_n)$ is not necessarily quasi-random.

We shall prove, among other things, that, for every regular sample graph $L_\nu$, $\nu\geqslant 4$, if the number of induced copies of $L_\nu$ in every induced $F_h\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^\nu)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\in (0,1)$, $e(L_\nu)>0$, and $L_\nu\ne K_\nu$).

We conjecture the same conclusion for every $L_\nu$ with $\nu\ge 4$, i.e., even if we drop the assumption of regularity.

We shall reduce the general problem to solving a system of polynomials. This gives a ‘simple’ algorithm to decide the problem for every given $L_\nu$.

We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].

For an integer $s\ges 2$, a property $\P^{(s)}$ is an infinite class of s-uniform hypergraphs closed under isomorphism. We say that a property $\P^{(s)}$ is \emph{hereditary\/} if~$\P^{(s)}$ is closed under taking induced subhypergraphs. Thus, for some `forbidden class' $\FF=\{\F_i^{(s)}\:i\in I\}$ of s-uniform hypergraphs, $\P^{(s)}$ is the set of all s-uniform hypergraphs not containing any $\F_i^{(s)}\in\FF$ as an induced subhypergraph. Let $\P^{(s)}_n$ be those hypergraphs of $\P^{(s)}$ on some fixed n-vertex set. For a set of s-uniform hypergraphs $\FF=\{\F_i^{(s)}\:i\in I\}$, let

\[ \exind(n,\FF)=\max\bigl|[n]^s{\setminus}(\M\cup\N)\vphantom{\big|}\bigr|, \]

where the maximum is taken over all $\M$ and $\N\subseteq[n]^s$ with $\M\cap\N=\emptyset$ such that, for all $\G\subseteq[n]^s{\setminus}(\M\cup\N)$, no $\F_i^{(s)}\in\FF$ appears as an induced subhypergraph of $\G\cup\M$. We show that

\[ \log_2\big|\P^{(3)}_n\big|=\exind(n,\FF)+o(n^3) \]

holds for $s=3$ and any hereditary property $\P^{(3)}$, where $\FF$ is a forbidden class associated with $\P^{(3)}$. This result complements a collection of analogous theorems already proved for graphs (i.e., $s=2$).

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $\mathbb{Z}$.

The number of spanning trees in the giant component of the random graph (n, c/n) (c > 1) grows like exp{m(f(c)+o(1))} as n → ∞, where m is the number of vertices in the giant component. The function f is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on f′(c). A key lemma is the following. Let PGW(λ) denote a Galton–Watson tree having Poisson offspring distribution with parameter λ. Suppose that λ*>λ>1. We show that PGW(λ*) conditioned to survive forever stochastically dominates PGW(λ) conditioned to survive forever.

This is a survey of a number of recent papers dealing with graphs from a geometric perspective. The main theme of these studies is the relationship between graph properties that are local in nature, and global graph parameters. Connections with the theory of distributed computing are pointed out and many open problems are presented.

A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.