Un -tableau d’Euler

In [R. Clarke, G.N. Han, J. Zeng, A combinatorial interpretation of the Seidel generation of q-derangement numbers, Ann. Comb. 1 (1997) 313–327] Clarke, Han and Zeng introduced a generalized Euler’s difference table. In this paper, we add a third variable and give a combinatorial interpretation of this generalization.     

d‐homogeneous and d‐ultrahomogeneous linear spaces

A linear space S is d‐homogeneous if, whenever the linear structures induced on two subsets S₁ and S₂ of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂. S is called d‐ultrahomogeneous if each isomorphism between the linear structures induced on two subsets of cardinality at most d can be extended into an automorphism of S. We have proved in [11;] (without any finiteness assumption) that every 6‐homogeneous linear space is homogeneous (that is d‐homogeneous for every positive integer d). Here we classify completely the finite nontrivial linear spaces that are d‐homogeneous for d ≥ 4 or d‐ultrahomogeneous for d ≥ 3. We also prove an existence theorem for infinite nontrivial 4‐ultrahomogeneous linear spaces. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 321–329, 2000     

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov () system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D(m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D(m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D(m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.     

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)

where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.     

Sur les extensions lineaires d'une famille particuliere d'ordres partiels

Corners are defined as ideals of an ordered integer half-dihedron; the paper develops a method of enumeration of the linear extensions of a given corner by means of an alternating sum of products of trinomials. The main result substantially generalizes previously known results and is by itself the starting point of generalizations to some further ordered sets.     

Hypergraphes de chaines d'aretes d'un arbre

We introduce the ‘edges-paths hypergraph of a tree’ and study relations of this notion with graphic geometries, chordable graphs. As particular case, we give a simple characterization of intervals hypergraphs.     

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

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

.     

The acyclic edge chromatic number of a random d‐regular graph is d + 1

We prove the theorem from the title: the acyclic edge chromatic number of a random d‐regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov, and Zaks and presents further support for a conjecture that Δ(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analog of a result of Robinson and the second author on edge chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 69–74, 2005     

Inexpensive d‐dimensional matchings

Suppose that independent U(0, 1) weights are assigned to the ${d\choose 2}n^{2}$ edges of the complete d‐partite graph with n vertices in each of the d maximal independent sets. Then the expected weight of the minimum‐weight perfect d‐dimensional matching is at least $\frac{3}{16}n^{1-(2/d)}$. We describe a randomized algorithm that finds a perfect d‐dimensional matching whose expected weight is at most 5d³n^1?(2/d)+d¹⁵ for all d≥3 and n≥1. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 50–58, 2002     

Partitions lineaires arguesiennes d'un espace vectoriel

In this work, we study linear and desarguesian partitions of a finite dimensional vector space over a skew-field K. When K is finite, we describe the set of all these partitions as a homogeneous space of the general linear group and we give an enumeration formula.     

Graphes cubiques d'indice trois,graphes cubiques isochromatiques,graphes cubiques d'indice quatre

The process introduced by E. Johnson [Amer. Math. Monthly73 (1966), 52–55] for constructing connected cubic graphs can be modified so as to obtain restricted classes of cubic graphs, in particular, those defined by their chromatic number or their chromatic index. We construct here the graphs of chromatic number three and the graphs whose chromatic number is equal to its chromatic index (isochromatic graphs). We then give results about the construction of the class of graphs of chromatic index four, and in particular, we construct an infinite class of “snarks.”     

Isomorphismes d'hypergraphes par intersections équicardinales d'arêtes et configurations exclues

In this note we characterize isomorphism between two hypergraphs by means of equicardinality of certain edge intersections and the exclusion of certain pairs of subhypergraphs. Our result is slightly stronger than Theorem 3 of C. Berge and R. Rado (J. Combinatorial Theory Ser. B13 (1972), 226–241) in particular in that isolated vertices are admitted. As a corollary we obtain a result due to J.-L. Paillet.     

Polytope des independants d'un graphe serie-parallele

A graph is said to be serie-parallel if it doesn't contain an homeomorph to K₄. The aim of the paper is the demonstration of Chvatal's conjecture on the polytope of independent set of vertices in such graphs. This is done classically by using LP-duality, the algorithm for constructing the primal-dual solution having the nice property to be linear in the number of vertices.     

On semi‐classical d‐orthogonal polynomials

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.     

Nombre chromatique total d'un hypergraphe

In this paper, the author defines a hypergraph total chromatic number and determines this number for the complete h-partite and for the complete h-uniform hypergraphs.     

Arbres minimaux d'un graphe preordonne

We study the minimal spanning trees of a connected graph G = (X,U) where U is partially preordered (or quasi-ordered). We characterize several kinds of optimal spanning trees and give conditions for existence of strongly optimal trees. Generalizations to bases of matroids (binary matroïds in part 2) are immediate. Sone of our results are given in terms of Krugdahl's dependence graphs. They imply previous results of Rosenstiehl and Gale in the case of linear orders or preorders.     

Écoulement d'un fluide de Bingham dans un domaine non-cylindrique

We obtain an existence theorem for Bingham flows in a noncylindrical domain, as a simple application of a previous result on Bingham flows with a monotone perturbation.