33 results on '"Univerzita Karlova"'
Search Results
2. EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Sequence ,Logic ,010102 general mathematics ,0102 computer and information sciences ,Characterization (mathematics) ,01 natural sciences ,Combinatorics ,Philosophy ,Definable set ,Monotone polygon ,Integer ,010201 computation theory & mathematics ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Standard probability space ,Limit (mathematics) ,Ball (mathematics) ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:1.If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.2.A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.
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- 2019
3. A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
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Jaroslav Nesetril, Patrice Ossona de Mendez, Computer Science Institute of Charles University [Prague] (IUUK), Charles University [Prague] (CU), Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Supported by grant ERCCZ LL-1201 and CE-ITI P202/12/G061, and by the European Associated Laboratory 'Structures in Combinatorics' (LEA STRUCO), Department of Applied Mathematics (KAM) (KAM), and Univerzita Karlova v Praze
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Model theory ,General Mathematics ,Stone space ,0102 computer and information sciences ,Tree-depth ,01 natural sciences ,Graph ,Combinatorics ,Definable set ,Measurable graph ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Mathematics - Combinatorics ,Radon measures ,Relational structure ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Connected component ,Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,Graph limits ,Colored ,010201 computation theory & mathematics ,Bounded function ,Standard probability space ,Combinatorics (math.CO) ,First-order logic ,Tuple ,Structural limits - Abstract
In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this put these into next context and perspective. For example, we prove that the sparse--dense dichotomy exactly corresponds to random free graphons. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limits objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role of elementary brick these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of {\em modeling} we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures., Comment: added journal reference
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- 2020
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4. Approximations of Mappings
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Combinatorics ,Model theory ,Discrete mathematics ,Conjecture ,Unary operation ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Function (mathematics) ,Limit (mathematics) ,Inverse problem ,Unary function ,ComputingMilieux_MISCELLANEOUS ,Decidability ,Mathematics - Abstract
We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem of the construction of a continuous limit for first-order convergent sequences of finite mappings. We solve the approximation problem and, consequently, the full characterization of limit objects for mappings for first-order (i.e. \(\mathrm{FO}\)) convergence and local (i.e. \(\mathrm{FO}^\mathrm{local}\)) convergence. This work can be seen both as a first step in the resolution of inverse problems (like Aldous–Lyons conjecture) and a strengthening of the classical decidability result for finite satisfiability in Rabin class (which consists of first-order logic with equality, one unary function, and an arbitrary number of monadic predicates). The proof involves model theory and analytic techniques.
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- 2019
5. First-Order Interpretations of Bounded Expansion Classes
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Ossona De Mendez, Patrice, Gajarsky, Jakub, Kreutzer, Stephan, Nešetřil, Jaroslav, Mendez, Patrice Ossona De, Pilipczuk, Michał, Siebertz, Sebastian, Toruńczyk, Szymon, Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Masaryk University [Brno] (MUNI), Technische Universität Berlin (TU), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Faculty of Mathematics, Informatics, and Mechanics [Warsaw] (MIMUW), University of Warsaw (UW), Universität Bremen, and Wagner, Michael
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FOS: Computer and information sciences ,Model checking ,Computer Science - Logic in Computer Science ,Discrete Mathematics (cs.DM) ,General Computer Science ,Logic ,Computer science ,0102 computer and information sciences ,Characterization (mathematics) ,01 natural sciences ,Theoretical Computer Science ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Bounded expansion ,Mathematics - Combinatorics ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Discrete mathematics ,000 Computer science, knowledge, general works ,010102 general mathematics ,Mathematics - Logic ,First order ,Graph ,Logic in Computer Science (cs.LO) ,First-order logic ,Computational Mathematics ,010201 computation theory & mathematics ,Computer Science ,Combinatorics (math.CO) ,Logic (math.LO) ,Computer Science - Discrete Mathematics - Abstract
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion , defined as first-order transductions of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth covers (or colorings), replacing treedepth by its dense analogue called shrubdepth.
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- 2018
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6. Nauczanie aspektu czasowników polskich na gruncie słowiańskim
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Maria Magdalena Nowakowska, Univerzita Karlova, Katedra Středoevropských Studií, and Uniwersytet Łódzki, Katedra Lingwistyki Stosowanej i Kulturowej
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technolekt ,Czech ,History ,teaching Polish as a foreign language ,First language ,General Engineering ,technolect ,nauczanie języka polskiego jako obcego ,Verb ,teaching Polish for specific purposes ,verb aspect ,language.human_language ,Linguistics ,lcsh:Philology. Linguistics ,lcsh:P1-1091 ,language ,Slovak ,Language for specific purposes ,Slavic languages ,aspekt czasownika ,nauczanie języka specjalistycznego - Abstract
This article shows the principle of teaching the verb aspect in Polish by means of examples taken from other Slavonic languages. It presents the differences among verb aspects found in Polish, Czech, Slovak and Slovenian. The illustrative examples also document the final numer of mistakes made by foreign students whose first languages miss the right equivalent of the verb aspect. Special attention has been paid to the importance of teaching the verbal aspect in classes of language for specific purposes. W artykule podjęto problematykę nauczania aspektu czasownika na gruncie słowiańskim. Zaprezentowano różnice, dotyczące tej kategorii językowej, w językach polskim, czeskim, słowackim i słoweńskim, a następnie pokazano wynikające z tego problemy w przyswajaniu tej kategorii przez Czechów, Słowaków i Słoweńców. Szczególną uwagę zwrócono na aspekt w nauczaniu języków specjalistycznych.
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- 2017
7. A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems
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Martin Vohralík, Vít Dolejší, Alexandre Ern, Department of Numerical Mathematics, Univerzita Karlova v Praze, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Environmental Modeling, Optimization and Programming Models (POMDAPI), Inria Paris-Rocquencourt, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
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Approximation property ,unified framework ,flux reconstruction ,discontinuous Galerkin method ,robustness ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Discontinuous Galerkin method ,Robustness (computer science) ,0101 mathematics ,Dual norm ,Mathematics ,a posteriori estimate ,dual norm ,Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Estimator ,unsteady nonlinear advection-diffusion problem ,flux equilibration ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,A priori and a posteriori ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; We derive a framework for a posteriori error estimates in unsteady, nonlinear, possibly degenerate, advection-diffusion problems. Our estimators are based on a space-time equilibrated flux reconstruction and are locally computable. They are derived for the error measured in a space-time mesh-dependent dual norm stemming from the problem and meshes at hand augmented by a jump seminorm measuring possible nonconformities in space. Owing to this choice, a guaranteed and globally efficient upper bound is achieved, as well as robustness with respect to nonlinearities, advection dominance, domain size, final time, and absolute and relative size of space and time steps. Local-in-time and in-space efficiency is also shown for a localized upper bound of the error measure. In order to apply the framework to a given numerical method, two simple conditions, local space-time mass conservation and an approximation property of the reconstructed fluxes, need to be verified. We show how to do this for the interior-penalty discontinuous Galerkin method in space and the Crank--Nicolson scheme in time. Numerical experiments illustrate the theory.
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- 2013
8. The fractional chromatic number of Zykov products of graphs
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Jean Sébastien Sereni, Pierre Charbit, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, and ANR-10-JCJC-0204,HEREDIA,Classes héréditaires de graphes(2010)
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Discrete mathematics ,Mathematics::Combinatorics ,Conjecture ,Recurrence relation ,Applied Mathematics ,Computer Science::Neural and Evolutionary Computation ,010102 general mathematics ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,01 natural sciences ,Combinatorics ,05C15 ,Corollary ,010201 computation theory & mathematics ,Product (mathematics) ,Chromatic scale ,Fractional coloring ,0101 mathematics ,Triangle-free graphs ,Zykov graphs ,Mathematics - Abstract
Zykov designed one of the oldest known families of triangle-free graphs with arbitrarily high chromatic number. We determine the fractional chromatic number of the Zykov product of a family of graphs. As a corollary, we deduce that the fractional chromatic numbers of the Zykov graphs satisfy the same recurrence relation as those of the Mycielski graphs, that is a n + 1 = a n + 1 a n . This solves a conjecture of Jacobs.
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- 2011
9. Edge-face coloring of plane graphs with maximum degree nine
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Matěj Stehlík, Jean-Sébastien Sereni, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Optimisation Combinatoire (G-SCOP_OC), Laboratoire des sciences pour la conception, l'optimisation et la production (G-SCOP), Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS), and Institut National Polytechnique de Grenoble (INPG)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut National Polytechnique de Grenoble (INPG)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)
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Discrete mathematics ,plane graph ,010102 general mathematics ,0102 computer and information sciences ,Complete coloring ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,1-planar graph ,Brooks' theorem ,Combinatorics ,Greedy coloring ,Edge coloring ,05C15 ,010201 computation theory & mathematics ,graph coloring ,edge-face coloring ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,Graph coloring ,0101 mathematics ,Fractional coloring ,Mathematics ,List coloring - Abstract
An edge-face coloring of a plane graph with edge set E and face set F is a coloring of the elements of E∪F so that adjacent or incident elements receive different colors. Borodin [Discrete Math 128(1–3):21–33, 1994] proved that every plane graph of maximum degree Δ⩾10 can be edge-face colored with Δ + 1 colors. We extend Borodin's result to the case where Δ = 9. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:332-346, 2011 © 2011 Wiley Periodicals, Inc.
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- 2010
10. Graphs with bounded tree-width and large odd-girth are almost bipartite
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Daniel Král, Jean-Sébastien Sereni, Alexandr V. Kostochka, Michael Stiebitz, Department of Mathematics [Urbana], University of Illinois at Urbana-Champaign [Urbana], University of Illinois System-University of Illinois System, Institute of Mathematics [Novosibirsk], Institute of Mathematics of Novosibirsk, Institut teoretické informatiky (ITI), Charles University [Prague] (CU), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics - Technical University of Ilmenau, and Ilmenau University of Technology [Germany] (TU)
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Odd-girth ,Circular coloring ,Tree-width ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Chromatic scale ,05C83 ,0101 mathematics ,Mathematics ,Discrete mathematics ,010102 general mathematics ,05C15 ,Graph ,Treewidth ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Bounded function ,Bipartite graph ,Combinatorics (math.CO) - Abstract
International audience; We prove that for every k and every \epsilon > 0, there exists g such that every graph with tree-width at most k and odd-girth at least g has circular chromatic number at most 2 + \epsilon.
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- 2010
11. Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs
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Edita Máčajová, Jean-Sébastien Sereni, Attila Pór, Daniel Král, Institut teoretické informatiky (ITI), Charles University [Prague] (CU), Department of Computer Science - Comenius University, Comenius University in Bratislava, Department of Mathematics - Western Kentucky University (WKU), Western Kentucky University (WKU), Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics (KAM) (KAM), and Univerzita Karlova v Praze
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Discrete mathematics ,Vertex (graph theory) ,General Mathematics ,010102 general mathematics ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,Edge (geometry) ,01 natural sciences ,Steiner tree problem ,Combinatorics ,symbols.namesake ,Steiner system ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0103 physical sciences ,symbols ,Cubic graph ,05B07, 05C15 ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective nonaffine point-transitive Steiner triple system S.
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- 2010
12. The Last Fraction of a Fractional Conjecture
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Jean-Sébastien Sereni, Daniel Král, František Kardoš, Institut teoretické informatiky (ITI), Charles University [Prague] (CU), Institute of Mathematics, P.J. Safarik University, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), and Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Graph ,girth ,Combinatorics ,05C15 ,total coloring ,Computer Science::Discrete Mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,fractional coloring ,Mathematics - Combinatorics ,05C15, 05C72 ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Abstract
Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most $\Delta+1+\varepsilon$. The conjecture was proven to be true when $\Delta=3$ or $\Delta$ is even. We settle the conjecture by proving it for the remaining cases., Comment: A typo has been corrected in the introduction (concerning the citation of the result by Ito, Kennedy and Reed)
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- 2010
13. Counting Homomorphisms to Sparse Graphs
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,Dense graph ,Applied Mathematics ,Nowhere dense set ,Tree-depth ,Graph ,Combinatorics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Discrete Mathematics and Combinatorics ,Homomorphism ,Graph homomorphism ,Boolean conjunctive query ,Mathematics - Abstract
International audience; We define nowhere dense and somewhere dense classes by means of counting of homomorphisms from test graphs. This seems to be bridging the gap between existential and counting theorems (for graph homomorphisms) and it has application to complexity of Boolean queries
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- 2009
14. Untangling a Planar Graph
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Andreas Spillner, Chan-Su Shin, Xavier Goaoc, Yoshio Okamoto, Alexander Wolff, Jan Kratochvíl, Effective Geometric Algorithms for Surfaces and Visibility (VEGAS), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Discrete Optimization Laboratory, Toyohashi University of Technology, Departement of Electronics and Information Engineering, Hankuk University of Foreign Studies, Faculteit Wiskunde & Informatica, Technische Universiteit Eindhoven (TU/e), Algorithms, and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
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Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,G.2.2 ,0102 computer and information sciences ,Computer Science::Computational Geometry ,[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG] ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,symbols.namesake ,Computer Science::Discrete Mathematics ,Outerplanar graph ,Discrete Mathematics and Combinatorics ,I.3.5 ,0101 mathematics ,Mathematics ,010102 general mathematics ,Binary logarithm ,Graph ,Planar graph ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,symbols ,Computer Science - Computational Geometry ,Geometry and Topology ,Computer Science - Discrete Mathematics - Abstract
A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to untangle $\delta$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,\delta)=n-shift(G,\delta)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $\delta$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $\delta_G$ of $G$ with fix$(G,\delta_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $\delta_H$ of $H$ with fix$(H,\delta_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs., Comment: (v5) Minor, mostly linguistic changes
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- 2009
15. A New Lower Bound on the Number of Perfect Matchings in Cubic Graphs
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Michael Stiebitz, Jean-Sébastien Sereni, Daniel Král, Institut teoretické informatiky (ITI), Charles University [Prague] (CU), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics - Technical University of Ilmenau, and Ilmenau University of Technology [Germany] (TU)
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Discrete mathematics ,perfect matching polytope ,cubic graph ,05C70 ,General Mathematics ,perfect matching ,010102 general mathematics ,Strong perfect graph theorem ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Upper and lower bounds ,Tutte theorem ,Graph ,brick and brace decomposition ,Combinatorics ,Computer Science::Discrete Mathematics ,010201 computation theory & mathematics ,Trivially perfect graph ,Cubic graph ,0101 mathematics ,Mathematics - Abstract
International audience; We prove that every n-vertex cubic bridgeless graph has at least n/2 perfect matchings and give a list of all 17 such graphs that have less than n/2+2 perfect matchings.
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- 2009
16. List colorings with measurable sets
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Jan Hladký, Daniel Kráal, Jean-Sébastien Sereni, Michael Stiebitz, Centre for Discrete Mathematics and its Applications [Warwick] (DIMAP), University of Warwick [Coventry], Institut teoretické informatiky (ITI), Charles University [Prague] (CU), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics - Technical University of Ilmenau, and Ilmenau University of Technology [Germany] (TU)
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list coloring ,fractional chromatic number ,010102 general mathematics ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,16. Peace & justice ,Hall's theorem ,01 natural sciences ,05C15 ,010201 computation theory & mathematics ,Discrete Mathematics and Combinatorics ,measurable sets ,Geometry and Topology ,0101 mathematics - Abstract
International audience; The measurable list chromatic number of a graph G is the smallest number such that if each vertex v of G is assigned a set L(v) of measure in a fixed atomless measure space, then there exist sets such that each c(v) has measure one and for every pair of adjacent vertices v and v'. We provide a simpler proof of a measurable generalization of Hall's theorem due to Hilton and Johnson [J Graph Theory 54 (2007), 179-193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number.
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- 2008
17. Grad and classes with bounded expansion III. Restricted graph homomorphism dualities
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Ossona de Mendez, Patrice, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,Algebra homomorphism ,Degree (graph theory) ,Context (language use) ,Hadwiger conjecture (graph theory) ,Theoretical Computer Science ,Combinatorics ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Computational Theory and Mathematics ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Discrete Mathematics and Combinatorics ,Graph homomorphism ,Homomorphism ,Geometry and Topology ,Induced homomorphism (fundamental group) ,Mathematics - Abstract
National audience; We study restricted homomorphism dualities in the context of classes with bounded expansion (which are defined by means of the greatest reduced average densities—grads). This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximate version” of the Hadwiger conjecture.
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- 2008
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18. Grad and classes with bounded expansion I. Decompositions
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,05C15 05C75 ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Bounded expansion ,Discrete Mathematics and Combinatorics ,Partition (number theory) ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Invariant (mathematics) ,Mathematics - Abstract
International audience; We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). For these classes we prove the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. This generalizes and simplifies several earlier results (obtained for minor closed classes).
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- 2008
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19. Grad and classes with bounded expansion II. Algorithmic aspects
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,Dense graph ,Subgraph isomorphism problem ,05C85 05C78 ,Graph theory ,Tree-depth ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Homomorphism ,Combinatorics (math.CO) ,Geometry and Topology ,Graph property ,Time complexity ,Mathematics - Abstract
Classes of graphs with bounded expansion have been introduced in [J. Nesetril, P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, O. Delmas (Eds.), 7th International Colloquium on Graph Theory, in: Electronic Notes in Discrete Mathematics, vol. 22, Elsevier (2005), pp. 101-106; J. Nesetril, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)]. They generalize classes with forbidden topological minors (i.e. classes of graphs having no subgraph isomorphic to the subdivision of some graph in a forbidden family), and hence both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N(C,p) so that the vertex set of any graph [email protected]?C may be partitioned into at most N(C,p) parts, any [email protected]?p parts of them induce a subgraph of tree-width at most (i-1) [J. Nesetril, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005) (submitted for publication)] (actually, of tree-depth [J. Nesetril, P. Ossona de Mendez, Tree depth, subgraph coloring and homomorphism bounds, European Journal of Combinatorics 27 (6) (2006) 1022-1041] at most i, which is sensibly stronger). Such partitions are central to the resolution of homomorphism problems like restricted homomorphism dualities [J. Nesetril, P. Ossona de Mendez, Grad and classes with bounded expansion III. Restricted dualities, European Journal of Combinatorics (2005) (submitted for publication)]. We give here a simple algorithm for computing such partitions and prove that if we restrict the input graph to some fixed class C with bounded expansion, the running time of the algorithm is bounded by a linear function of the order of the graph (for fixed C and p). This result is applied to get a linear time algorithm for the subgraph isomorphism problem with fixed pattern and input graphs in a fixed class with bounded expansion. More generally, let @f be a first-order logic sentence. We prove that any fixed graph property of type may be decided in linear time for input graphs in a fixed class with bounded expansion. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. Also, extending several earlier results, we show that a class of graphs has sublinear separators if it has sub-exponential expansion. This result is best possible in general.
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- 2008
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20. Total-Coloring of Plane Graphs with Maximum Degree Nine
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Jean-Sébastien Sereni, Łukasz Kowalik, Riste Škrekovski, Institute of Informatics [Warsaw], Faculty of Mathematics, Informatics, and Mechanics [Warsaw] (MIMUW), University of Warsaw (UW)-University of Warsaw (UW), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Faculty of Mathematics and Physics [Ljubljana] (FMF), and University of Ljubljana
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Discrete mathematics ,total-coloring ,Conjecture ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Discharging method ,planar graph ,Total coloring ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,16. Peace & justice ,01 natural sciences ,Graph ,Planar graph ,Combinatorics ,symbols.namesake ,05C15 ,010201 computation theory & mathematics ,symbols ,discharging method ,0101 mathematics ,Mathematics - Abstract
The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree $\Delta$ admits a $(\Delta+2)$-total-coloring. Similar to edge-colorings—with Vizing's edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if $\Delta\ge10$, then every plane graph of maximum degree $\Delta$ is $(\Delta+1)$-totally-colorable. On the other hand, such a statement does not hold if $\Delta\le3$. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.
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- 2008
21. Fraternal Augmentations of graphs, Coloration and Minors
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), and Elsevier
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Discrete mathematics ,Applied Mathematics ,Subgraph isomorphism problem ,Tree-depth ,1-planar graph ,Combinatorics ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Discrete Mathematics and Combinatorics ,Graph homomorphism ,Induced subgraph isomorphism problem ,Homomorphism ,Graph isomorphism ,Mathematics - Abstract
We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r , ∇ r ( G ) . We generalize to these classes some results proved for proper minor closed classes and bounded degree graphs, such as the existence of low tree-width colorings, a linear time algorithm to check subgraph isomorphism for a fixed pattern and homomorphism dualities.
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- 2007
22. Counterweight Continent, Dehydrated Ocean and Carrack Mountain – Welcome to T. Pratchett’s Discworld
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Žaneta Dvořáková, Gałkowski, Artur, Gliwa, Renata, and Univerzita Karlova v Praze
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toponimy ,funkcje onimiczne ,onomastika literacka - Abstract
Terry Pratchett has created a fictional world, an integral part of which is a detailed and sophisticated system of place names. In his novels there are realistic names giving us an illusion of the real world and often serving as a characterization as well. Association is typical for a further kind of literary toponyms, the names refer to actual places on Earth, they activate our general cultural knowledge encyclopedia and play with the reader and his/her interpretive abilities. The last group of toponyms is purely artistic, they were created as a language game, probably the most important criterion was their aesthetic appeal. The names often deliberately destroy the illusion of reality and probability (we can talk about the anti-illusionist function), they demonstrate their artificiality, that they are a part of the author’s game. Whatever the motivation, meaning or possible allusion, all toponyms in the fictional world of the novels operate in the same way. They locate the story and help in spacial orientation within the Discworld, the names are an integral part of how the characters perceive and experience their space. Udostępnienie publikacji Wydawnictwa Uniwersytetu Łódzkiego finansowane w ramach projektu „Doskonałość naukowa kluczem do doskonałości kształcenia”. Projekt realizowany jest ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Wiedza Edukacja Rozwój; nr umowy: POWER.03.05.00-00-Z092/17-00.
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- 2015
23. Cuts and bounds
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,05C15,06F99 ,Degree (graph theory) ,Homomorphism order ,Cuts ,Hadwiger conjecture (graph theory) ,Infimum and supremum ,Theoretical Computer Science ,Combinatorics ,Bounds ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Homomorphism ,Order (group theory) ,Discrete Mathematics and Combinatorics ,Homomorphism extrema ,Finite set ,Maximal element ,Mathematics - Abstract
We consider the colouring (or homomorphism) order C induced by all finite graphs and the existence of a homomorphism between them. This ordering may be seen as a lattice which is far from being complete. In this paper we study bounds and suprema and maximal elements in C of some frequently studied classes of graphs (such as bounded degree, degenerated and classes determined by a finite set of forbidden subgraphs). We relate these extrema to cuts of subclasses K of C (cuts are finite sets which are comparable to every element of the class K). We determine all cuts for classes of degenerated graphs. For classes of bounded degree graphs this seems to be a very difficult problem which is also mirrored by the fact that these classes fail to have a supremum. We note a striking difference between undirected and oriented graphs. This is based on the recent work of C. Tardif and J. Nešetřil. Also minor closed classes are considered and we survey recent results obtained by authors. A bit surprisingly this order setting captures Hadwiger conjecture and suggests some new problems.
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- 2005
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24. On the maximum average degree and the oriented chromatic number of a graph
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Alexandr V. Kostochka, Eric Sopena, Oleg V. Borodin, Jaroslav Nešetřil, André Raspaud, Raspaud, André, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire Bordelais de Recherche en Informatique (LaBRI), and Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)
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Discrete mathematics ,Mathematics::Combinatorics ,010102 general mathematics ,Oriented coloring ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Butterfly graph ,Simplex graph ,Theoretical Computer Science ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Windmill graph ,010201 computation theory & mathematics ,Graph power ,Computer Science::Discrete Mathematics ,Friendship graph ,Petersen graph ,Wheel graph ,Physics::Accelerator Physics ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
The oriented chromatic number o( H ) of an oriented graph H is defined as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′. The oriented chromatic number o( G ) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o( G ) and mad( G ) defined as the maximum average degree of the subgraphs of G.
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- 1999
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25. Educational Policies and Inequalities in Europe
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Daniel Frandji, Jean-Yves Rochex, Marc Demeuse, David Greger, Institut d'Administration scolaire (INAS), Université de Mons (UMons), Triangle : action, discours, pensée politique et économique (TRIANGLE), École normale supérieure - Lyon (ENS Lyon)-Université Lumière - Lyon 2 (UL2)-Sciences Po Lyon - Institut d'études politiques de Lyon (IEP Lyon), Université de Lyon-Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Pedagogicka fakulta (PEDF), Univerzita Karlova v Praze, Centre interdisciplinaire de recherche, culture, éducation, formation, travail (CIRCEFT), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Sciences Po Lyon - Institut d'études politiques de Lyon (IEP Lyon), Université de Lyon-Université de Lyon-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and Frandji, Daniel
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Czech ,Economic growth ,Index (economics) ,Inequality ,[SHS.SOCIO] Humanities and Social Sciences/Sociology ,politiques d'éducation prioritaire ,public policy ,inégalités scolaires ,[SHS.EDU]Humanities and Social Sciences/Education ,media_common.quotation_subject ,[SHS.EDU] Humanities and Social Sciences/Education ,Psychological intervention ,lutte contre l'exclusion ,Decentralization ,0504 sociology ,inequalities ,Political science ,parcours scolaires ,Education policy ,media_common ,education ,[SHS.SOCIO]Humanities and Social Sciences/Sociology ,Equity (economics) ,EuroPEP ,4. Education ,05 social sciences ,1. No poverty ,050401 social sciences methods ,050301 education ,internationale comparison ,[SHS.SCIPO]Humanities and Social Sciences/Political science ,Democracy ,language.human_language ,Europe ,language ,Justice in education ,[SHS.SCIPO] Humanities and Social Sciences/Political science ,0503 education - Abstract
List of Tables Acknowledgements Foreword S.Power Introduction: Towards a Comparison of Priority Education Policies in Europe D.Frandji Policy Interventions to Reduce Educational Inequalities: The Case of England L.Antoniou, A.Dyson & C.Raffo Priority Education Policies in Belgium: Two Modes of Regulation of the Effects of a Market Logic N.Friant, M.Demeuse, A.Aubert-Lotarski & I.Nicaise Twenty Five Years of Priority Education Policy in France: Dubious Specificity and Disappointing Results J.Rochex Greece: Educational Programmes in Support and Innovation G.Varnava-Skoura, D.Vergidis & C.Kassimi From the Invention of the Democratic City to the Management of Exclusion and Urban Violence in Portugal J.Correia, I.Cruz, J.Rochex & L.Salgado Priority Education Policies in the Czech Republic: Redesigning Equity Policies in the Post-Communist Transformation D.Greger , M.Levinska & I.Smetackova Romania: A System in Evolution, Searching for its Conceptual References C.Rus Sweden: Priority Education Policies in Times of Decentralization and Individualization G.Francia & L.Herrera Conclusion: Priority Education Policies in Europe: From One 'Age' and One Country to Another J.Rochex Bibliography Index
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- 2012
26. Colorings and Homomorphisms of Minor Closed Classes
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Jaroslav Nešetřil, Patrice Ossona de Mendez, Ossona de Mendez, Patrice, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,homomorphism ,Mathematics::Combinatorics ,Hadwiger conjecture (graph theory) ,Chromatic polynomial ,Butterfly graph ,Planar graph ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,symbols.namesake ,05C15 ,Windmill graph ,Computer Science::Discrete Mathematics ,Friendship graph ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Triangle-free graph ,graph coloring ,symbols ,Acyclic orientation ,minor closed class ,Mathematics - Abstract
We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. This solves a problem posed in [[15]]. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.
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- 2003
27. Graphs with full rank 3-color matrix and few 3-colorings
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Jean-Sébastien Sereni, Serguei Norine, Department of Mathematics - Princeton University, Princeton University, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), and Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
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Discrete mathematics ,Vector space ,Rank (linear algebra) ,010102 general mathematics ,Incidence matrix ,0102 computer and information sciences ,Vertex coloring ,Computer Science::Computational Geometry ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Distance-regular graph ,Graph ,Theoretical Computer Science ,Combinatorics ,05C15 ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Graph power ,Triangle-free graph ,Discrete Mathematics and Combinatorics ,Graph homomorphism ,Adjacency matrix ,Graph coloring ,0101 mathematics ,Mathematics - Abstract
International audience; We exhibit a counterexample to a conjecture of Thomassen stating that the number of distinct 3-colorings of every graph whose 3-color matrix has full column rank is superpolynomial in the number of vertices.
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- 2008
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28. Pattern avoidance in partial permutations (extended abstract)
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Sergey Kitaev, Vít Jelínek, Anders Claesson, Eva Jelínková, The Mathematics Institute, Reyjavik University (School of Computer Science), Reykjavík University, Institute of Mathematics (Reykjavik University), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Billey, Sara and Reiner, and Victor
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Wilf-equivalence ,Sequence ,Mathematics::Combinatorics ,General Computer Science ,Partial permutation ,Baxter permutation ,Generating function ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,Theoretical Computer Science ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Permutation ,pattern avoidance ,generating function ,bijection ,partial permutation ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Bijection ,Discrete Mathematics and Combinatorics ,Equivalence class ,Equivalence (measure theory) ,Mathematics - Abstract
Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes''. We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n \geq k \geq 1$., Nous introduisons un concept de permutations partielles. $\textit{Une permutation partielle de longueur n avec k trous}$ est une suite finie de symboles $\pi = \pi_1 \pi_2 \cdots \pi_n$ dans laquelle chaque nombre de l'ensemble $\{1,2,\ldots,n-k\}$ apparaît précisément une fois, tandis que les $k$ autres symboles de $\pi$ sont des "trous''. Nous introduisons l'étude des permutations partielles à motifs exclus et nous montrons que la plupart des résultats sur l'équivalence de Wilf peuvent être généralisés aux permutations partielles avec un nombre arbitraire de trous. De plus, nous montrons que les permutations de Baxter d'une longueur donnée $k$ forment une classe d'équivalence du type Wilf par rapport aux permutations partielles avec $(k-2)$ trous. Enfin, nous présentons l'énumération des permutations partielles de longueur $n$ avec $k$ trous qui évitent un motif de longueur $\ell \leq 4$, pour chaque $n \geq k \geq 1$.
29. Colorings and girth of oriented planar graphs
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André Raspaud, Eric Sopena, Jarik Nešetřil, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), and Raspaud, André
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Discrete mathematics ,Mathematics::Combinatorics ,Book embedding ,010102 general mathematics ,0102 computer and information sciences ,Girth (graph theory) ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Indifference graph ,Pathwidth ,Computer Science::Discrete Mathematics ,010201 computation theory & mathematics ,Chordal graph ,Outerplanar graph ,Triangle-free graph ,Odd graph ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even if we insist the girth of G is large, an assumption which makes the chromatic number easy to compute. In particular, we prove that every orientation of any large girth planar graph is Scolorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth oriented planar graphs. 1. Introduction and statement of results Given graphs G = (V, E) and G’ = (V’, E’) a homomorphism from G to G’ is any mapping f : V -+ V’ satisfying LGYI E E*[f(x),f(v)l E E’. Here the brackets on both sides of the implication means the same thing: either an edge or an arc. The existence of a homomorphism from G to G’ will be denoted by G + G’. Homomorphisms are clearly related to the chromatic number of undirected graphs (an undirected graph G is k-colorable if and only if there exists a homomorphism from G to
30. Fast Exact Algorithm for L(2,1)-Labeling of Graphs
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Peter Rossmanith, Mathieu Liedloff, Konstanty Junosza-Szaniawski, Pawel Rzazewski, Jan Kratochvíl, Faculty of Mathematics and Information Science [Warszawa], Warsaw University of Technology [Warsaw], Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Laboratoire d'Informatique Fondamentale d'Orléans (LIFO), Ecole Nationale Supérieure d'Ingénieurs de Bourges-Université d'Orléans (UO), Department of Computer Science [Aachen], and Rheinisch-Westfälische Technische Hochschule Aachen (RWTH)
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Discrete mathematics ,Graph labeling ,Edge-graceful labeling ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,Neighbourhood (graph theory) ,0102 computer and information sciences ,02 engineering and technology ,16. Peace & justice ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,Graph power ,0202 electrical engineering, electronic engineering, information engineering ,Wheel graph ,020201 artificial intelligence & image processing ,Path graph ,Graph homomorphism ,ComputingMilieux_MISCELLANEOUS ,Complement graph ,Mathematics - Abstract
International audience; An L(2,1)-labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. The span of such a labeling is the maximum label used, and the L(2,1)-span of a graph is the minimum possible span of its L(2,1)-labelings. We show how to compute the L(2,1)-span of a connected graph in time $O*(2.6488^n)$. Previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3.2361, with 3 seemingly having been the Holy Grail.
31. PCR La confluence Saône-Doubs à l'âge du Fer (Vie s. av. J.-C. au Ier siècle de notre ère): Annexes
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Émilie Dubreucq, Matthieu Thivet, David Bardel, Philippe Barral, Marion Berranger, Veronica Cicolani, Frédéric Cruz, Hélène Duchamp, Benjamin Dufour, Émilien Estur, Anne Filippini, Jean-Loup Flouest, Morgane Guillot, Stéphane Izri, Luc Jaccottey, Régis Labeaune, Caroline Lachiche, Matthieu Le Bailly, Nicolas Monot, Pierre Nouvel, Fabienne. Olmer, Pierre Péfau, Maxence Pieters, Joëlle Rolland, Réjane Roure, Federica Sachetti, Christelle Sanchez, Caroline Schaal, Julien Soichet, Valérie Taillandier, Quentin Verriez, Grégory Videau, Travaux et recherches archéologiques sur les cultures, les espaces et les sociétés (TRACES), Ministère de la Culture et de la Communication (MCC)-École des hautes études en sciences sociales (EHESS)-Université Toulouse - Jean Jaurès (UT2J)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Chrono-environnement - CNRS - UBFC (UMR 6249) (LCE), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), UMR 6249 Chrono-environnement, UMR 5608 TRACES, Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Institut national de recherches archéologiques préventives (Inrap), Université Bourgogne Franche-Comté [COMUE] (UBFC), Centre National de la Recherche Scientifique (CNRS), Université de Technologie Belfort-Montbéliard (UTBM), IRAMAT - Laboratoire Métallurgies et Cultures (IRAMAT - LMC), Institut de Recherches sur les Archéomatériaux (IRAMAT), Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Montaigne-Université de Technologie de Belfort-Montbeliard (UTBM)-Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Montaigne-Université de Technologie de Belfort-Montbeliard (UTBM), Archéologie et Philologie d'Orient et d'Occident (AOROC), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Ghent Archaeological Team, BIBRACTE EPCC, Mission Archéologique Départementale de l'Eure (MADE), Service Départemental de l'Archéologie, EVEHA (Etudes et valorisations archeologiques), Archéologie, Terre, Histoire, Sociétés [Dijon] (ARTeHiS), Centre National de la Recherche Scientifique (CNRS)-Université de Bourgogne (UB)-Ministère de la Culture et de la Communication (MCC), Université de Bourgogne (UB), Centre Camille Jullian - Histoire et archéologie de la Méditerranée et de l'Afrique du Nord de la protohistoire à la fin de l'Antiquité (CCJ), Aix Marseille Université (AMU)-Ministère de la Culture et de la Communication (MCC)-Centre National de la Recherche Scientifique (CNRS), Centre ardennais de recherche archéologique (CARA), Univerzita Karlova v Praze, Université Paul-Valéry - Montpellier 3 (UPVM), Archéologie des Sociétés Méditerranéennes (ASM), Université Paul-Valéry - Montpellier 3 (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Ministère de la Culture (MC), Service Régional d'Archéologie PACA, Ministère de la Culture et de la Communication (MCC), GeoArchEon SARL, École des hautes études en sciences sociales (EHESS)-Université Toulouse - Jean Jaurès (UT2J)-Ministère de la Culture et de la Communication (MCC)-Centre National de la Recherche Scientifique (CNRS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL), and Ministère de la Culture et de la Communication (MCC)-Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)
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Archélogie ,Archéométrie ,[SHS.ARCHEO]Humanities and Social Sciences/Archaeology and Prehistory ,Télédétection ,Bioarcheologie ,Culture Matérielle ,Analyse documentaire ,Géophysiques ,Paléoenvironnements - Abstract
Ce rapport est un document administratif destiné à rendre compte des actions réalisées au cours de l’année 2019 dans le cadre du Projet Collectif de Recherche consacré à la confluence Saône-Doubs à l’âge du Fer, coordonné par Emilie Dubreucq et Matthieu Thivet
32. Equivalences for pattern avoiding involutions and classification
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Astrid Reifegerste, Mark Dukes, Toufik Mansour, Vít Jelínek, Science Institute, University of Iceland, University of Iceland [Reykjavik], Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Department of Mathematics [Haïfa], University of Haifa [Haifa], Faculty of Mathematics, Otto-von-Guericke University [Magdeburg] (OVGU), Krattenthaler, Christian and Sagan, and Bruce
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Discrete mathematics ,pattern avoiding involutions ,Mathematics::Combinatorics ,General Computer Science ,pattern avoiding permutations ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,16. Peace & justice ,Wilf equivalence ,Theoretical Computer Science ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,forbidden subsequences ,signed permutations ,Symmetric group ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Discrete Mathematics and Combinatorics ,Mathematics - Abstract
We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions., Nous complétons la classification de Wilf des motifs signés de longueur 5 à la fois pour les permutations signées et les involutions signées. Nous donnons de nouvelles équivalences générales de motifs qui prouvent les conjectures de Jaggard concernant les involutions dans le groupe symétrique évitant certains motifs de longueur 5 et 6. De cette manière nous complétons également la classification de Wilf de $S_5$, $S_6$ et $S_7$ à la fois pour les permutations et les involutions.
33. Sparse Combinatorial Structures: Classification and Applications
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Patrice Ossona de Mendez, Jaroslav Nesetril, Ossona de Mendez, Patrice, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
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Model theory ,Model checking ,Mathematical logic ,Discrete mathematics ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Nowhere dense set ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Homomorphism ,Graph theory ,Context (language use) ,Extremal graph theory ,Mathematics - Abstract
International audience; We present results of the recent research on sparse graphs and finite structures in the context of of contemporary combinatorics, graph theory, model theory and mathematical logic, complexity of algorithms and probability theory. The topics include: complexity of subgraph- and homomorphism- problems; model checking problems for first order formulas in special classes; property testing in sparse classes of structures. All these problems can be studied under the umbrella of classes of structures which are Nowhere Dense and in the context of Nowhere Dense -- Somewhere Dense dichotomy. This dichotomy presents the classification of the general classes of structures which proves to be very robust and stable as it can be defined alternatively by most combinatorial extremal invariants as well as by algorithmic and logical terms. We give examples from logic, geometry and extremal graph theory. Finally we characterize the existence of all restricted dualities in terms of limit objects defined on the homomorphism order of graphs.
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