On hamiltonian colorings of block graphs

A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that $D(u,v)+|c(u)-c(v)|\geq p-1$ for every two distinct vertices u and v of G, where D(u,v) denoted the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs.Comment: 12 pages, 1 figure. A conference version appeared in the proceedings of WALCOM 201

Conflict-free connection numbers of line graphs

A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is a conflict-free path connecting them. For a connected graph $G$, the \emph{conflict-free connection number} of $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required to make $G$ conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We first show that for an arbitrary connected graph $G$, there exists a positive integer $k$ such that $cfc(L^k(G))\leq 2$. Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph $G$ and an arbitrary positive integer $k$, we always have $cfc(L^{k+1}(G))\leq cfc(L^k(G))$, with only the exception that $G$ is isomorphic to a star of order at least~$5$ and $k=1$. Finally, we obtain the exact values of $cfc(L^k(G))$, and use them as an efficient tool to get the smallest nonnegative integer $k_0$ such that $cfc(L^{k_0}(G))=2$.Comment: 11 page

Les composants structurels du discours argumentatif écrit selon un modèle construit à des fins didactiques pour la classe de français

Inspirée par une lecture nouvelle de la Rhétorique d'Aristote et par les recherches récentes dans les sciences du langage, l'auteure a construit un modèle systémique du discours argumentatif écrit, préalable à l'élaboration d'une didactique de l'argumentation pour la classe de français (secondaire et collégial). Cet article présente une partie des propriétés structurelles de ce type de discours en définissant ses composants essentiels et propose une conception différente de celle qui prévaut actuellement dans l'enseignement du discours argumentatif.Based on a renewed reading of Artistotie's Rhetoric and on recent research in the area of language, the author constructs a systemic model of written argumentative discourse which is seen as preliminary to the development of a didactic process of argumentative reasoning in french class (at secondary and college levels). This article presents a portion of the structural properties of this type of discourse by defining the essential components and then proposes an alternative view to that currently described in the teaching of argumentative discours.Inspirada por una nueva lectura de la Retôrica de Aristoteles y por las investigaciones recientes en las ciencias del lenguaje, la autora construyo un modelo sistémico del discurso argumentative escrito, previo a la elaboracion de una didactica de la argumentacion para el curso de francés (secundario y pre-universitario). Este articulo présenta una parte de las propiedades estructurales de este tipo de discurso, al définir sus componentes esenciales y propone una concepeion alternativa a la que prevalece actualemente en la ensefianza del discurso argumentative.Angeregt von einer neuerlichen Lektiire der Rhetorik des Aristoteles und durch jiingste Forschungen in den SprachwifJenschaften, hat die Verfaperin ein systemisches Modell der schriftlichen Beweisfiihrung gebaut, als Vorspann fur die Erarbeitung einer Didaktik der Beweisfûrung fiir den Franzosischunterricht (an der hôheren Schule und der Kollegstufe). Dieser Artikel legt einen Teil der strukturellen Eigenschaften dieser Art der Rede vor, indem er seine wesentlichen Komponenten definiert, und schlâgt eine Alternative zu der Auffapung vor, die gegenwârtig beim Unterrichten der Beweisrede vorherrscht

Compétences à mobiliser pour la compréhension et l’interprétation de manuels d’histoire du secondaire au Québec

Une analyse de la mise en discours de deux chapitres de deux manuels d’histoire générale conçus pour des élèves québécois de 13-14 ans a été menée afin de connaître quelles compétences ces élèves doivent mobiliser en lecture pour que ces manuels constituent un réel outil d’apprentissage, a fortiori lorsqu’ils se présentent comme des substituts de l’enseignant. Il en ressort que, sans un étayage systématique de la part des enseignants pour assurer la compréhension et l’interprétation des élèves, ces manuels ne peuvent jouer leur rôle, car ils présentent de trop nombreux obstacles pour de jeunes lecteurs.This article presents an analysis of the writing in two chapters in two general history textbooks for Quebec students aged 13-14 years old. The objective was to determine the reading competencies required to use these texts as a real learning tool when presented as a teacher substitute. The results show that, without a systematic presentation by teachers for ensuring student comprehension and interpretation, these textbooks cannot serve their role, since there are too many obstacles for young readers.Se llevó a cabo un análisis del discurso de dos capítulos de dos libros de texto en historia general, concebidos para alumnos quebequenses de 13-14 años para conocer las competencias que estos alumnos deben de movilizar en lectura para que estos libros de texto constituyan una verdadera herramienta de aprendizaje, con más razón cuando pretenden ser sustitutos del docente. Se destaca que, si los docentes no proporcionan sistemáticamente más información a los alumnos para asegurarse de su comprensión y de su interpretación, estos libros de texto no pueden cumplir con las expectativas dado que presentan demasiados obstáculos para los jóvenes lectores

On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao

Depleted pyrochlore antiferromagnets

I consider the class of "depleted pyrochlore" lattices of corner-sharing triangles, made by removing spins from a pyrochlore lattice such that every tetrahedron loses exactly one. Previously known examples are the "hyperkagome" and "kagome staircase". I give criteria in terms of loops for whether a given depleted lattice can order analogous to the kagome \sqrt{3} \times \sqrt{three} state, and also show how the pseudo-dipolar correlations (due to local constraints) generalize to even the random depleted case.Comment: 6pp IOP latex, 1 figure; Proc. "Highly Frustrated Magnetism 2008", Sept 2008, Braunschwei