Improving upon work in [2], the precise value of the set reconstruction number is given for all cyclic groups of odd order.

Une substitution est un morphisme de monoïdes libres :chaque lettre a pour image un mot, etl'image d'un mot est laconcaténation des images de ses lettres.Cet article introduit unegénéralisation de la notion de substitution,où l'image d'une lettre n'est plus un mot mais un motif, c'est-à-direun “mot à trous”, l'image d'un mot étant obtenue en raccordant lesmotifs correspondant à chacune de ses lettres à l'aide de règleslocales. On caractérisecomplètement les substitutions par des motifs qui sontdéfinies sur toute suite biinfinie, et on explique comment lesconstruire. On montre que toutesuite biinfinie qui est point fixe d'une substitution par des motifsest substitutive, c'est-à-dire est l'image, par un morphisme lettre àlettre, d'un point fixe de substitution (au sens usuel).

We study some arithmetical and combinatorial properties ofβ-integers for β being the larger root of the equationx2 = mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3. We determine withthe accuracy of ± 1 the maximal number of β-fractionalpositions, which may arise as a result of addition of twoβ-integers. For the infinite word uβ> coding distancesbetween the consecutive β-integers, we determine preciselyalso the balance. The word uβ> is the only fixed point of themorphism A → Am-1B and B → Am-n-1B. In the case n = 1,the corresponding infinite word uβ> is sturmian, and,therefore, 1-balanced. On the simplest non-sturmian example withn≥ 2, we illustrate how closely the balance and thearithmetical properties of β-integers are related.

We explore the borderline between decidability and undecidability of the following question: “Let C be a class of codes. Given a machine ${\mathfrak{M}}$ of type X, is it decidable whether the language $L({{\mathfrak{M}}})$ lies in C or not?” for codes in general, ω-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix codes, and for finite automata equipped with different versions of push-down stores and counters.

This paper has two parts. In the first part we consider a simple Markov chain for d-regular graphs on n vertices, where d = d(n) may grow with n. We show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. In the second part of the paper, a related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even constant d. This is a model for a certain peer-to-peer network. We prove that the related chain has mixing time which is bounded above by a polynomial in N, the expected number of vertices, provided certain assumptions are met about the rate of arrival and departure of vertices.

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node, without leaving any trace. We consider the task of locating a black hole in a (partially) synchronous tree network, assuming an upper bound on the time of any edge traversal by an agent. The minimum number of agents capable of identifying a black hole is two. For a given tree and given starting node we are interested in the fastest-possible black hole search by two agents. For arbitrary trees we give a 5/3-approximation algorithm for this problem. We give optimal black hole search algorithms for two ‘extreme’ classes of trees: the class of lines and the class of trees in which any internal node (including the root which is the starting node) has at least two children.

We consider relations between thresholds for monotone set properties and simple lower bounds for such thresholds. A motivating example (Conjecture 2): Given an n-vertex graph H, write pE for the least p such that, for each subgraph H' of H, the expected number of copies of H' in G=G(n, p) is at least 1, and pc for that p for which the probability that G contains (a copy of) H is 1/2. Then (conjecture) pc=O(pElog n). Possible connections with discrete isoperimetry are also discussed.

The chromatic polynomial PΓ(x) of a graph Γ is a polynomial whose value at the positive integer k is the number of proper k-colourings of Γ. If G is a group of automorphisms of Γ, then there is a polynomial OPΓ,G(x), whose value at the positive integer k is the number of orbits of G on proper k-colourings of Γ.

It is known that real chromatic roots cannot be negative, but they are dense in ∞). Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in , but under certain hypotheses, there are zero-free regions.

We also look at orbital flow roots. Here things are more complicated because the orbit count is given by a multivariate polynomial; but it has a natural univariate specialization, and we show that the roots of these polynomials are dense in the negative real axis.

Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic.

We show that almost surely , where G is a random tree of order n or the giant component of a random graph with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.

In a previous paper we showed that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6-regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random d-regular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right-hand sides.

Consider the set of finite words on a totally ordered alphabet with two letters. We prove that the distribution of the length of the standard right factor of a random Lyndon word with length n, divided by n, converges towhen n goes to infinity. The convergence of all moments follows. This paper thus completes the results of [2], in which the limit of the first moment is given.

We show that in the game of angel and devil, played on the planar integer lattice, the angel of power 4 can evade the devil. This answers a question of Berlekamp, Conway and Guy. Independent proofs that work for the angel of power 2 have been given by Kloster and by Máthé.

We solve Conway's Angel Problem by showing that the Angel of power 2 has a winning strategy.

An old observation of Conway is that we may suppose without loss of generality that the Angel never jumps to a square where he could have already landed at a previous time. We turn this observation around and prove that we may suppose without loss of generality that the Devil never eats a square where the Angel could have already jumped. Then we give a simple winning strategy for the Angel.

Let G1 and G2 be graphs of order n with maximum degree Δ1 and Δ2, respectively. G1 and G2 are said to pack if there exist injective mappings of the vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer showed that if , then G1 and G2 pack. We extend this result by showing that if , then G1 and G2 do not pack if and only if one of G1 or G2 is a perfect matching and the other either is with odd or contains .

Let d ≥ d0 be a sufficiently large constant. An graph G is a d-regular graph over n vertices whose second-largest (in absolute value) eigenvalue is at most . For any 0<p<1, Gp is the graph induced by retaining each edge of G with probability p. It is known that for the graph Gp almost surely contains a unique giant component (a connected component with linear number vertices). We show that for the giant component of Gp almost surely has an edge expansion of at least .

We show that the sampling formula induced from a Λ-coalescent process with multiple collisions is regenerative if and only if the measure Λ is either concentrated in 0 (Kingman case) or concentrated in 1 (star-shaped case). The Ewens sampling formula is the only sampling formula in this class which also belongs to Pitman's two-parameter family of sampling distributions.