We prove that the chromatic polynomial of a finite graph of maximal degree Δ is free of zeros for |q| ≥ C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

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).

We study two topological properties of the 5-ary n-cube $Q_{n}^{5}$ . Given two arbitrary distinct nodes x and y in $Q_{n}^{5}$ , we prove that there exists anx-y path of every length ranging from 2n to 5n - 1, where n ≥ 2. Basedon this result, we prove that $Q_{n}^{5}$ is5-edge-pancyclic by showing that every edge in $Q_{n}^{5}$ lies ona cycle of every length ranging from 5 to 5n .

We survey several quantitative problems on infinite words relatedto repetitions, recurrence, and palindromes, for which the Fibonacciword often exhibits extremal behaviour.

We first prove an extremal propertyof the infiniteFibonacci*word f: the family of the palindromic prefixes{hn | n ≥ 6}of fis not only a circular code but “almost” a comma-free one(see Prop. 12 in Sect. 4).We also extend to a more general situationthe notion of a necklace introducedfor the study of trinucleotides codes on the genetic alphabet,and we present a hierarchyrelating two important classes of codes,the comma-free codes and the circular ones.

We discuss some known and introduce some new hierarchies andreducibilities on regular languages, with the emphasis on thequantifier-alternation and difference hierarchies of thequasi-aperiodic languages. The non-collapse of these hierarchies anddecidability of some levels are established. Complete sets in thelevels of the hierarchies under the polylogtime and somequantifier-free reducibilities are found. Some facts about thecorresponding degree structures are established. As an application,we characterize the regular languages whose balanced leaf-languageclasses are contained in the polynomial hierarchy. For anydiscussed reducibility we try to give motivations and openquestions, in a hope to convince the reader that the study of thesereducibilities is interesting for automata theory and computationalcomplexity.

In this paper, we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A*, then the right and left ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from a nonempty word. We show that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, we prove that for any given pseudostandard word s with seed, all sufficiently long left special factors of s are prefixes of it.

Let $\mathcal{L}$ be a language.A balanced pair (u,v) consists of two words u and v in $\mathcal{L}$ which have the same number ofoccurrences of each letter.It is irreducible if the pairs of strict prefixes of u and v of the same length do not form balanced pairs. In this article, we are interested in computing the set of irreducible balanced pairs on several cases of languages.We make connections with the balanced pairs algorithm and discrete geometrical constructions related to substitutive languages.We characterize substitutive languages which have infinitely many irreducible balanced pairs of a given form.

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizesunivoque real numbers; the other is a disguised version of the set of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic Sturmian sequence beginning itself in 1.

In this paper, we characterize the substitutions over athree-letter alphabet which generate a ultimately periodicsequence.

The LS (Look and Say) derivative of a word is obtained by writing the number of consecutive equal letters when the word is spelled from left to right. For example, LS( 1 1 2 3 3) = 2 1 1 2 2 3 (two 1, one 2, two 3). We start the study of the behaviour of binary words generated by morphisms under the LS operator, focusing in particular on the Fibonacci word.

In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function f has at least one point of continuity and that its continuity set C(f) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational ${\bf \Pi}^0_2$ -subset of Σω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σω .

This paper is a contribution to the general tiling problem for the hyperbolic plane.It is an intermediary result between the result obtained by R. Robinson [Invent. Math.44 (1978) 259–264]and the conjecture that the problem is undecidable.

We design algorithms of “optimal" data complexity for several natural problems about first-order queries on structures of bounded degree. For that purpose, we first introduce a framework to deal with logical or combinatorial problems R ⊂ I x O whose instances x ∈ I may admit of several solutions R(x) = {y ∈ O : (x,y) ∈ R}. One associates to such a problem several specific tasks: compute a random (for the uniform probability distribution) solution y ∈ R(x); enumerate without repetition each solution y j in some specific linear order y0 < y1 < ... < yn-1 where R(x) = {y0,...,yn-1}; compute the solution y j of rankj in the linear order <.Algorithms of “minimal" data complexity are presented for the following problems: given any first-order formula $\varphi(\bar{v})$ and any structure S of bounded degree:(1) compute a random element of $\varphi(S)=\{\bar{a}: (S,\bar{a})\models\varphi(\bar{v})\}$ ;(2) compute the jth element of $\varphi(S)$ for some linear order of $\varphi(S)$ ;(3) enumerate the elements of $\varphi(S)$ in lexicographical order.More precisely, we prove that, for any fixed formula φ, the above problem (1) (resp. (2), (3)) can be computed within average constant time (resp. within constant time, with constant delay) after a linear (O(|S|)) precomputation. Our essential tool for deriving those complexity results is a normalization procedure of first-order formulas on bijective structures.

Here is presented a 6-states non minimal-time solution which is intrinsically Minsky-like and solves the three following problems: unrestricted version on a line, with one initiator at each end of a line and the problem on a ring. We also give a complete proof of correctness of our solution, which was never done in a publication for Minsky's solutions.

Given m positive integers R = (ri), n positive integers C = (cj) such that Σri = Σcj = N, and mn non-negative weights W=(wij), we consider the total weight T=T(R, C; W) of non-negative integer matrices D=(dij) with the row sums ri, column sums cj, and the weight of D equal to . For different choices of R, C, and W, the quantity T(R,C; W) specializes to the permanent of a matrix, the number of contingency tables with prescribed margins, and the number of integer feasible flows in a network. We present a randomized algorithm whose complexity is polynomial in N and which computes a number T′=T′(R,C;W) such that T′ ≤ T ≤ α(R,C)T′ where . In many cases, ln T′ provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N × N random matrix with exponentially distributed entries and approximate the expectation by the integral T′ of an efficiently computable log-concave function on ℝmn.

Let Γ =(V,E) be a point-symmetric reflexive relation and let υ ∈ V such that |Γ(υ)| is finite (and hence |Γ(x)| is finite for all x, by the transitive action of the group of automorphisms). Let j ∈ℕ be an integer such that Γj(υ)∩ Γ−(υ)={υ}. Our main result states that

As an application we have |Γj(υ)| ≥ 1+(|Γ(υ)|−1)j. The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta–Häggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.

We define the class of discrete classical categorial grammars, similar inthe spirit to the notion of reversible class of languages introduced by Angluin andSakakibara. We show that the class of discrete classical categorial grammars is identifiable from positive structured examples. For this, we provide an original algorithm, which runs in quadratic time in the size of the examples. This work extends the previous results of Kanazawa. Indeed, in our work, several types can be associated to a word and the class is still identifiable in polynomial time. We illustrate the relevance of the class of discrete classical categorial grammars with linguistic examples.