We investigate the number of 4-edge paths in graphs with a given number of vertices and edges, proving an asymptotically sharp upper bound on this number. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.

The depth of a trie has been deeply studied when the source which produces the words is a simple source (a memoryless source or a Markov chain). When a source is simple but not an unbiased memoryless source, the expectation and the variance are both of logarithmic order and their dominant terms involve characteristic objects of the source, for instance the entropy. Moreover, there is an asymptotic Gaussian law, even though the speed of convergence towards the Gaussian law has not yet been precisely estimated. The present paper describes a ‘natural’ class of general sources, which does not contain any simple source, where the depth of a random trie, built on a set of words independently drawn from the source, has the same type of probabilistic behaviour as for simple sources: the expectation and the variance are both of logarithmic order and there is an asymptotic Gaussian law. There are precise asymptotic expansions for the expectation and the variance, and the speed of convergence toward the Gaussian law is optimal. The paper first provides analytical conditions on the Dirichlet series of probabilities of a general source under which this Gaussian law can be derived: a pole-free region where the series is of polynomial growth. In a second step, the paper focuses on sources associated with dynamical systems, called dynamical sources, where the Dirichlet series of probabilities is expressed with the transfer operator of the dynamical system. Then, the paper extends results due to Dolgopyat, already generalized by Baladi and Vallée, and shows that the previous analytical conditions are fulfilled for ‘most’ dynamical sources, provided that they ‘strongly differ’ from simple sources. Finally, the present paper describes a class of sources not containing any simple source, where the trie depth has the same type of probabilistic behaviour as for simple sources, even with more precise estimates.

We prove an essentially sharp $\tilde \Omega (n/k)$ lower bound on the k-round distributional complexity of the k-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson’s $\tilde \Omega (n/{k^2})$ lower bound, and essentially matches the randomized lower bound proved by Klauck. The proof is information-theoretic, and a key part of it is using asymmetric triangular discrimination instead of total variation distance; this idea may be useful elsewhere.

Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of Kr such that no Kr is rainbow.

We consider numbers and sizes of independent sets in graphs with minimum degree at least d. In particular, we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.

We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in $\mathbb{R}^d$ is (1 ± o(1)) $\frac{(2d-2)!}{(d-1)!}$n lnd−1n, if d is fixed. This estimate is relevant to analysis of the performance of exact algorithms for computing the largest empty axis-parallel box amidst n given points in an axis-parallel box R, especially the algorithms that proceed by examining all maximal empty boxes. Our method for bounding the expected number of maximal empty boxes also shows that the expected number of maximal empty orthants determined by n random points in $\mathbb{R}^d$ is (1 ± o(1)) lnd−1n, if d is fixed. This estimate is related to the expected number of maximal (or minimal) points amidst random points, and has application to algorithms for coloured orthogonal range counting.

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt(n,p), defined to be the minimum number of edges that a Kp-saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt(n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt(n,p).

Let P be a set of n points in ℝ3, and let k ≤ n be an integer. A sphere σ is k-rich with respect to P if |σ ∩ P| ≥ k, and is η-non-degenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P| points of P.

We improve the previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ3, from O(n4/k5 + n3/k3), which holds for all 0 < η < 1/2, to O*(n4/k11/2 + n2/k2), which holds for all 0 < η < 1 (in both bounds, the constants of proportionality depend on η). The new bound implies the improved upper bound O*(n58/27) ≈ O(n2.1482) on the number of mutually similar triangles spanned by n points in ℝ3; the previous bound was O(n13/6) ≈ O(n2.1667) [1].

Dubhashi, Jonasson and Ranjan Dubhashi, Jonasson and Ranjan (2007) study the negative dependence properties of Srinivasan's sampling processes (SSPs), random processes which sample sets of a fixed size with prescribed marginals. In particular they prove that linear SSPs have conditional negative association, by using the Feder–Mihail theorem and a coupling argument. We consider a broader class of SSPs that we call tournament SSPs (TSSPs). These have a tree-like structure and we prove that they have conditional negative association. Our approach is completely different from that of Dubhashi, Jonasson and Ranjan. We give an abstract characterization of TSSPs, and use this to deduce that certain conditioned TSSPs are themselves TSSPs. We show that TSSPs have negative association, and hence conditional negative association. We also give an example of an SSP that does not have negative association.

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

A graph G is called n-minimizable if it can be reduced, by deleting a set of its edges, to a minimally n-connected graph. It is shown that, if n-connected graphs G and H differ only by finitely many vertices and edges, then G is n-minimizable if and only if H is n-minimizable (Theorem 4.12). In the main result, conditions are given that a tree decomposition of an n-connected graph G must satisfy in order to guarantee that the n-minimizability of each of the members of this decomposition implies the n-minimizability of the graph G (Theorem 6.5).

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.

A digraph is m-labelled if every arc is labelled by an integer in {1, . . ., m}. Motivated by wavelength assignment for multicasts in optical networks, we introduce and study n-fibre colourings of labelled digraphs. These are colourings of the arcs of D such that at each vertex v, and for each colour α, in(v, α) + out(v, α) ≤ n with in(v, α) the number of arcs coloured α entering v and out(v, α) the number of labels l such that there is at least one arc of label l leaving v and coloured with α. The problem is to find the minimum number of colours λn(D) such that the m-labelled digraph D has an n-fibre colouring. In the particular case when D is 1-labelled, λ1(D) is called the directed star arboricity of D, and is denoted by dst(D). We first show that dst(D) ≤ 2Δ−(D)+1, and conjecture that if Δ−(D) ≥ 2, then dst(D) ≤ 2Δ−(D). We also prove that for a subcubic digraph D, then dst(D) ≤ 3, and that if Δ+(D), Δ−(D) ≤ 2, then dst(D) ≤ 4. Finally, we study λn(m, k) = max{λn(D) | D is m-labelled and Δ−(D) ≤ k}. We show that if m ≥ n, then for some constant C. We conjecture that the lower bound should be the correct value of λn(m, k).

Let f be a de Morgan read-once function of n variables. Let fε be the random restriction obtained by independently assigning to each variable of f, the value 0 with probability (1 -ε)/2, the value 1 with the same probability, and leaving it unassigned with probability ε. We show that fε depends, on the average, on only O(εαn + εn1/α) variables, where . This result is asymptotically the tightest possible. It improves a similar result obtained recently by Håstad, Razborov and Yao.

Stacks which allow elements to be pushed into any of the top r positions and popped from any of the top s positions are studied. An asymptotic formula for the number un of permutations of length n sortable by such a stack is found in the cases r=1 or s=1. This formula is found from the generating function of un. The sortable permutations are characterized if r=1 or s=1 or r=s=2 by a forbidden subsequence condition.

In this paper we study distances in random subgraphs of a generalized n-cube [Qscr ]ns over a finite alphabet S of size s. [Qscr ]ns is the direct product of complete graphs over s vertices, its vertices being the n-tuples (x1, …, xn), with xi ∈ S, i = 1, … n, and two vertices being adjacent if they differ in exactly one coordinate. A random (induced) subgraph γ of [Qscr ]ns is obtained by selecting [Qscr ]ns-vertices with independent probability pn and then inducing the corresponding edges from [Qscr ]ns. Our main result is that dγ (P,Q) [les ] [2k+3]d[Qscr ]ns (P,Q) almost surely for P,Q ∈ γ, pn = n−a and 0 [les ] a < ½, where k = [1+3a/1−2a] and dγ and d[Qscr ]ns denote the distances in γ and [Qscr ]ns, respectively.

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph G0 in the common random graph models ${\cal G}$(n,m) and ${\cal G}$(n,p). Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of G0. This extends an argument given earlier by the second author for G0=K3 with a more restricted range of average degree. For all strictly balanced subgraphs G0, our results give much information on the distribution of the number of copies of G0 that are not in large ‘clusters’ of copies. The probability that a random graph in ${\cal G}$(n,p) has no copies of G0 is shown to be given asymptotically by the exponential of a power series in n and p, over a fairly wide range of p. A corresponding result is also given for ${\cal G}$(n,m), which gives an asymptotic formula for the number of graphs with n vertices, m edges and no copies of G0, for the applicable range of m. An example is given, computing the asymptotic probability that a random graph has no triangles for p=o(n−7/11) in ${\cal G}$(n,p) and for m=o(n15/11) in ${\cal G}$(n,m), extending results of the second author.