scholarly journals Relaxed Two-Coloring of Cubic Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Robert Berke ◽  
Tibor Szabó
Keyword(s):  
Regular Graph ◽  
Maximum Degree ◽  
Independent Set ◽  
The Other ◽  
Cubic Graphs ◽  
Other Hand ◽  
Color Class ◽  
International Audience

International audience We show that any graph of maximum degree at most $3$ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most $189$. On the other hand for any constant $C$ we exhibit a $4$-regular graph, such that the deletion of any independent set leaves at least one component of order greater than $C$. Similar results are obtained for coloring graphs of given maximum degree with $k+ \ell$ colors such that $k$ parts form an independent set and $\ell$ parts span components of order bounded by a constant. A lot of interesting questions remain open.

scholarly journals Large independent sets in triangle-free cubic graphs: beyond planarity

2020 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Gwenaël Joret ◽  
Jan Goedgebeur
Keyword(s):  
Planar Graph ◽  
Present Article ◽  
Additional Assumption ◽  
Independence Number ◽  
Independent Set ◽  
Cubic Graphs ◽  
Subcubic Graph ◽  
Color Class ◽  
Subcubic Graphs ◽  
Colorable Graph

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

scholarly journals Triangular fully packed loop configurations of excess 2

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Sabine Beil
Keyword(s):  
Boundary Conditions ◽  
Young Diagram ◽  
Rotational Invariance ◽  
The Other ◽  
Necessary Condition ◽  
Grand Nombre ◽  
Other Hand ◽  
International Audience ◽  
Linear Expression ◽  
Link Pattern

International audience Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions. A necessary condition for the boundary $(u,v;w)$ of a TFPL is $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, where $\lambda(u)$ denotes the Young diagram associated with the $01$-word $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. Later, Wieland drift was defined as the natural adaption of Wieland gyration to TFPLs. The main contribution of this article is a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$ in terms of numbers of stable TFPLs that is TFPLs invariant under Wieland drift. These stable TFPLs have boundary $(u^{+},v^{+};w)$ for words $u^{+}$ and $v^{+}$ such that $\lambda (u) \subseteq \lambda (u^{+})$ and $\lambda (v) \subseteq \lambda (v^{+})$. Les configurations de boucles compactes triangulaires (”triangular fully packed loop configurations”, ou TFPLs) sont apparues dans l’étude des configurations de boucles compactes dans un carré (FPLs) correspondant à des motifs de liaison avec un grand nombre d’arcs imbriqués. À chaque TPFL on associe un triplet $(u,v;w)$ de mots sur {0,1}, qui encode ses conditions aux bords. Une condition nécessaire pour le bord $(u,v;w)$ d’un TFPL est $\lvert \lambda(u) \rvert +\lvert \lambda(v) \rvert \leq \lvert \lambda(w) \rvert$, où $\lambda(u)$ désigne le diagramme de Young associé au mot $u$. D’un autre côté, la giration de Wieland a été inventée pour montrer l’invariance par rotation des nombres $A_\pi$ de FPLs correspondant à un motif de liaison donné $\pi$. Plus tard, la déviation de Wieland a été définie pour adapter de manière naturelle la giration de Wieland aux TFPLs. La contribution principale de cet article est une expression linéaire pour le nombre de TFPLs de bord $(u,v;w)$, où $\lvert \lambda (w) \rvert - \lvert\lambda (u) \rvert - \lvert \lambda (v)\rvert \leq 2$, en fonction des nombres de TFPLs stables, <i>i.e</i>., les TFPLs invariants par déviation de Wieland. Ces TFPLs stables ont pour bord $(u^{+},v^{+};w)$, avec $u^{+}$ et $v^{+}$ des mots tels que $\lambda (u) \subseteq \lambda (u^{+})$ et $\lambda (v) \subseteq \lambda (v^{+})$.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

More’s and Elyot’s Perspectives and their Classical Antecedents

Moreana ◽  
2003 ◽  
Vol 40 (Number 153- (1-2) ◽  
pp. 120-142 ◽  
Author(s):  
Howard B. Norland
Keyword(s):  
Social Problems ◽  
Social Values ◽  
Middle Level ◽  
Personal Relationship ◽  
The Other ◽  
Political Leaders ◽  
Social Construct ◽  
Other Hand ◽  
International Audience ◽  
Similarities And Differences

The nature of the personal relationship between More and Elyot remains something of a mystery, but their political perspectives reflect the similarities and differences in their public experience and sources of inspiration. More’s Utopia (1516) grew out of his interaction with European scholars and political leaders; composed in Latin for an international audience, the work moves from particular social problems in England to a radical alternative conceptualized in detail. The Governor (1531), written in English and addressed to the aristocracy, resulted from Elyot’s years as a middle-level public administrator. Practical rather than imaginative, Elyot finds the principles of governance mainly in Aristotle and Cicero; on the other hand, More, inspired primarily by Plato and Lucian, creates a social construct against which contemporary societies should be measured. Each reflecting his own perspective, Elyot emphasizes individual, personal virtues while More focuses on social values. The conservative Elyot endorses the hierarchal society governed by a monarch through his governors in contrast with More, who represents an elective meritocracy.

scholarly journals Sturmian Sequences and Invertible Substitutions

2011 ◽  
Vol Vol. 13 no. 2 (Combinatorics) ◽  
Author(s):  
Li Peng ◽  
Bo Tan
Keyword(s):  
Fixed Point ◽  
The Other ◽  
Sturmian Sequence ◽  
Other Hand ◽  
Angle Alpha ◽  
International Audience ◽  
Sturmian Sequences ◽  
Irrational Angle

Combinatorics International audience It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.

scholarly journals Is my paper relevant for an international audience?

2020 ◽  
Vol 18 (2) ◽  
pp. 1924
Author(s):  
Fernando Fernandez-Llimos
Keyword(s):  
The Other ◽  
Small Hospital ◽  
Cross Border ◽  
Other Hand ◽  
International Audience ◽  
Study Designs

This is the first question one should consider before submitting a paper to an international journal. The answer is simple: If researchers or practitioners from another country can learn something from your paper that can influence a practice or a research they are involved in, then your paper is relevant for an international audience. There are many elements that can influence in this cross-border transferability. One could think that having a big “n”, or performing complex statistical calculations, or using complicated study designs makes the paper more attractive to colleagues from other countries. These elements can help, but they are not sufficient. On the other hand, one could think that a study performed in a small hospital in a given country will never be of interest for these foreign colleagues. That is not necessarily correct. Let’s burst some myths.

scholarly journals Spanning paths in hypercubes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Tomáš Dvořák ◽  
Petr Gregor ◽  
Václav Koubek
Keyword(s):  
The Other ◽  
Other Hand ◽  
International Audience

International audience Given a family $\{u_i,v_i\}_{i=1}^k$ of pairwise distinct vertices of the $n$-dimensional hypercube $Q_n$ such that the distance of $u_i$ and $v_i$ is odd and $k \leq n-1$, there exists a family $\{P_i\}_{i=1}^k$ of paths such that $u_i$ and $v_i$ are the endvertices of $P_i$ and $\{V(P_i)\}_{i=1}^k$ partitions $V(Q_n)$. This holds for any $n \geq 2$ with one exception in the case when $n=k+1=4$. On the other hand, for any $n \geq 3$ there exist $n$ pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices.

scholarly journals Diameters and geodesic properties of generalizations of the associahedron

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
C. Ceballos ◽  
T. Manneville ◽  
V. Pilaud ◽  
L. Pournin
Keyword(s):  
The Other ◽  
Cluster Algebras ◽  
Other Hand ◽  
International Audience ◽  
The One ◽  
Generalized Associahedra

International audience The $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. L’associaèdre de dimension $n$ est un polytope dont les sommets correspondent aux triangulations d’un $(n + 3)$-gone convexe et dont les arêtes sont les échanges entre ces triangulations. Il a été récemment démontré que le diamètre de ce polytope est $2n - 4$ dès que $n > 9$. Nous étudions les diamètres des graphes de certaines généralisations de l’associaèdre : d’une part les associaèdres généralisés provenant des algèbres amassées, et d’autre part les associaèdres de graphes et les nestoèdres. En lien avec le diamètre, nous étudions si toutes les géodésiques entre deux sommets de ces polytopes restent dans la plus petite face les contenant.

scholarly journals Signed tree associahedra

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Vincent Pilaud
Keyword(s):  
Convex Polygon ◽  
The Other ◽  
Other Hand ◽  
International Audience ◽  
The One

International audience An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. J.-L. Loday gave a particularly elegant realization of the associahedron, which was then generalized in two directions: on the one hand to obtain realizations of graph associahedra, and on the other hand to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and extend these two constructions to signed tree associahedra. Un associaèdre est un polytope dont les sommets correspondent aux triangulations d’un polygone convexe et dont les arêtes correspondent aux flips entre ces triangulations. J.-L. Loday a donné une construction particulièrement élégante de l’associaèdre qui a été généralisée dans deux directions : d’une part pour obtenir des réalisations des associaèdres de graphes, et d’autre part pour obtenir de multiples réalisations de l’associaèdre paramétrées par une suite de signes. L’objectif de ce travail est d’unifier et d’étendre ces constructions aux associaèdres d’arbres signés.

scholarly journals Excluded subposets in the Boolean lattice

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gyula O.H. Katona
Keyword(s):  
Upper Bound ◽  
Boolean Lattice ◽  
The Other ◽  
Other Hand ◽  
International Audience ◽  
Element Set

International audience We are looking for the maximum number of subsets of an n-element set not containing 4 distinct subsets satisfying $A ⊂B, C ⊂B, C ⊂D$. It is proved that this number is at least the number of the $\lfloor \frac{n }{ 2}\rfloor$ -element sets times $1+\frac{2}{ n}$, on the other hand an upper bound is given with 4 replaced by the value 2.

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