Une generalisation du theoreme d'extension de Radó

Using subharmonic functions we give an easy proof of E.L. Stout's generalization of Radó's extension theorem.

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Travail partiellement subventionné par le Conseil National de Recherches du Canada (A 7668).     

Sur les parcours hamiltoniens dans les graphes orientes

We prove that in a strongly h-connected digraph of order n ? 8h³ + h there exists an Hamiltonian tour of length not exceeding

$\text{Max}\text{n+h}\text{2h}\text{n-h}\text{n-h}\text{2h}\text{,}\text{n+h}\text{2h}{}^1\text{n-h}\text{n-h}\text{2}{}^1$

. This bound is best possible.     

Theorie des ecoulements dans les graphes orientes sans circuit

The paper deals with tournaments (i.e., with trichotomic relations) and their homomorphisms. The study of tournaments by means of their homomorphisms is natural as tournaments are algebras of a special kind. We prove (1) theorems which relate combinatorial and algebraic notions (e.g., the score of a tournament and the monoid of its endomorphisms); (2) theorems concerned with strictly algebraic aspects of tournaments (e.g., characterizing the lattice of congruences of a tournament). Our main result is that the group of automorphisms and the lattice of congruences of a tournament are in general independent. In the last part of the paper we give some examples and applications to other fields.     

Theories egalitaires dans les Langages sur types de graphes

Sans résumé Ce travail doit son origine et son développement, aux préoccupations de recherche qui animent le groupe de “Logique et théorie des catégories,” dirigé par le Professeur A. Preller à l'U.E.R. de Luminy (Université d'Aix-Marseille II). Presented by J. D. Monk.     

Parcours dans les graphes: Un outil pour l'algorithmique des ensembles ordonnes

A finite poset can easily be represented by a directed acyclic graph. This work intends to make use of performant graph search methods as a tool for checking order properties. Semi-modular posets and semilattices are investigated here.An underlying idea consists in turning to account structural properties of the poset, and deriving effective algorithms. This purpose leads us to ‘good’ theoretic characterizations — that is, directly available from an algorithmic point of view — and especially a new one for modular lattices.     

Un theoreme sur les homomorphismes de schemas abeliens

Sans résumé À André Weil, pour son anniversaire     

Un theoreme sur les sommes amalgamees de groupes simpliciaux

In [1], D.M. Kan and R.F. Thurston prove two theorems on amalgamed sums of simplicial groups (cf corollaries 1 and 2 above) starting from a topological theorem of J.H.C. Whitehead. The proofs given in this paper are simplicial from the beginning; in the case where all groups are concentrated in dimension 0, one finds back J.H.C. Whitehead's theorem.     

Sur la separation dans les graphes de Cayley abeliens

In this note, we prove that Cayley digraphs on Abelian groups have some separation properties similar to undirected graphs.     

Sur un theoreme de separation des matroïdes orientes de rang trois

We prove a separation theorem for oriented matroids of rank 3. We give short combinatorial proofs of Carathéodory's theorem for arrangements of pseudohyperplanes and Kirchberger's theorem for arrangements of pseudolines.     

Une inegalite du type de Slepian et Gordon sur les processus gaussiens

A new proof and extension of the Slepian-Gordon inequality is given.      

Une involution sur les chemins de Dyck

We give a bijection between some valuated complete binary trees according to the number of leaves and multichains on a partially ordered set.     

Une remarque sur les sommes de Ramanujan

We study the asymptotic behaviour of the function $A(q)={\sup }_{x,y}|{\sum }_{x<n?y}{c}_q(n)|$ where $c_q(n)$ denotes the Ramanujan sum. We prove that $q?A(q)?q\sqrt{{\log }_{2}q}$ and determine how sharp this estimation is. To cite this article: A. Chadozeau, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Preuve ≪Elementaire≫ du theoreme de Grothendieck-Lefschetz pour les hypersurfaces

Sans résumé     

Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe oriente

In this article we prove that a sufficient condition for an oriented strongly connected graph with n vertices to be Hamiltonian is: (1) for any two nonadjacent vertices x and y

$\text{d}{}^{\mathrm{+}}\text{(x)+d}{}^{\mathrm{?}}\text{(x)+d}{}^{\mathrm{+}}\text{(y)+d}{}^{\mathrm{?}}\text{(y)?sn?1}$

.     

Aspects topologiques de la séparation dans les graphes infinis. II

Sans résumé     

Complexité et circuits eulériens dans les sommes tensorielles de graphes

The paper starts from two general remarks about the spectra of tensorial products and tensorial sums of square matrices. Since some classical results about spanning trees and Eulerian circuits may be reformulated in terms of eigenvalues of (what is called here) the second associated matrix, the initial remark about tensorial sums of matrices leads to natural applications to the enumeration of spanning trees and Eulerian circuits in tensorial sums (sometimes simply called “products”) of graphs. Examples and numerical results are given, in particular for the complexity of tensorial sums of chains and/or cycles and for the enumeration of the Eulerian circuits of the tensorial sum of two circuits.     

A propos d'une conjecture de F. Dejean sur les répétitions dans les mots

We show that with a four letter alphabet the largest unavoidable repetitions in arbitrarily long words are of the form uvu where the length of uvu is $\text{7}\text{5}$th of the length of uv. This proves part of a conjecture of F. Dejean.