MD  Matemàtica Discreta
http://hdl.handle.net/2117/3546
Fri, 27 Apr 2018 06:50:29 GMT
20180427T06:50:29Z

On the number of labeled graphs of bounded treewidth
http://hdl.handle.net/2117/115956
On the number of labeled graphs of bounded treewidth
Baste, Julien; Noy Serrano, Marcos; Sau, Ignasi
Let be Tnk the number of labeled graphs on vertices and treewidth at most (equivalently, the number o<f labeled partial trees). We show that [...] for k>1 and some explicit absolute constant c>0. Disregarding terms depending only on k, the gap between the lower and upper bound is of order (log k)n. The upper bound is a direct consequence of the wellknown formula for the number of labeled lambdatrees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and properpathwidth at most k .
Wed, 04 Apr 2018 14:01:07 GMT
http://hdl.handle.net/2117/115956
20180404T14:01:07Z
Baste, Julien
Noy Serrano, Marcos
Sau, Ignasi
Let be Tnk the number of labeled graphs on vertices and treewidth at most (equivalently, the number o<f labeled partial trees). We show that [...] for k>1 and some explicit absolute constant c>0. Disregarding terms depending only on k, the gap between the lower and upper bound is of order (log k)n. The upper bound is a direct consequence of the wellknown formula for the number of labeled lambdatrees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and properpathwidth at most k .

El arte de contar: combinatoria y enumeración
http://hdl.handle.net/2117/115600
El arte de contar: combinatoria y enumeración
Rué Perna, Juan José
Muchas de las preguntas más importantes de las matemáticas modernas requieren dominar un arte muy especial: el de contar. La rama de las matemáticas que ha hecho del enumerar un arte se llama combinatoria, y de la mano de figuras legendarias como Paul Erdös ha sido el marco de algunos de los resultados matemáticos más asombrosos del nuevo milenio
Fri, 23 Mar 2018 11:10:05 GMT
http://hdl.handle.net/2117/115600
20180323T11:10:05Z
Rué Perna, Juan José
Muchas de las preguntas más importantes de las matemáticas modernas requieren dominar un arte muy especial: el de contar. La rama de las matemáticas que ha hecho del enumerar un arte se llama combinatoria, y de la mano de figuras legendarias como Paul Erdös ha sido el marco de algunos de los resultados matemáticos más asombrosos del nuevo milenio

Los números trascendentes
http://hdl.handle.net/2117/115598
Los números trascendentes
Fresán, Javier; Rué Perna, Juan José
La expresión e¿ v163 es mucho más que la suma de sus partes e, ¿ y v163. Lejos de ser una elección casual, esta fórmula sirve a los autores de hilo conductor para adentrase en las áreas de la investigación más activa de la teoría de los números. De la mano de gigantes como Leonhard Euler, Pierre de Fermat o Évariste Galois, el lector emprenderá un viaje por la geometría aritmética que lo levara a explorar territorios tan dispares como las curvas elípticas, los periodos y las formas modulares. Y en el camino, como si de una novela policiaca se tratase, las vidas de estos objetos y de quienes los estudiaron se entrelazarán para resolver un misterio que ha fascinado a generaciones enteras de matemáticos ¿Por qué el número e¿ v163 está tan cercano a un numero entero?
Fri, 23 Mar 2018 10:58:07 GMT
http://hdl.handle.net/2117/115598
20180323T10:58:07Z
Fresán, Javier
Rué Perna, Juan José
La expresión e¿ v163 es mucho más que la suma de sus partes e, ¿ y v163. Lejos de ser una elección casual, esta fórmula sirve a los autores de hilo conductor para adentrase en las áreas de la investigación más activa de la teoría de los números. De la mano de gigantes como Leonhard Euler, Pierre de Fermat o Évariste Galois, el lector emprenderá un viaje por la geometría aritmética que lo levara a explorar territorios tan dispares como las curvas elípticas, los periodos y las formas modulares. Y en el camino, como si de una novela policiaca se tratase, las vidas de estos objetos y de quienes los estudiaron se entrelazarán para resolver un misterio que ha fascinado a generaciones enteras de matemáticos ¿Por qué el número e¿ v163 está tan cercano a un numero entero?

Transformation and decomposition of clutters into matroids
http://hdl.handle.net/2117/112728
Transformation and decomposition of clutters into matroids
Martí Farré, Jaume; Mier Vinué, Anna de
A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite nonempty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.
Fri, 12 Jan 2018 13:52:45 GMT
http://hdl.handle.net/2117/112728
20180112T13:52:45Z
Martí Farré, Jaume
Mier Vinué, Anna de
A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite nonempty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.

On trees with the same restricted Upolynomial and the Prouhet–Tarry–Escott problem
http://hdl.handle.net/2117/112678
On trees with the same restricted Upolynomial and the Prouhet–Tarry–Escott problem
Aliste Prieto, José; Mier Vinué, Anna de; Zamora, José
This paper focuses on the wellknown problem due to Stanley of whether two nonisomorphic trees can have the same Upolynomial (or, equivalently, the same chromatic symmetric function). We consider the Ukpolynomial, which is a restricted version of Upolynomial, and construct, for any given kk, nonisomorphic trees with the same Ukpolynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the Upolynomial up to isomorphism.
Thu, 11 Jan 2018 14:04:18 GMT
http://hdl.handle.net/2117/112678
20180111T14:04:18Z
Aliste Prieto, José
Mier Vinué, Anna de
Zamora, José
This paper focuses on the wellknown problem due to Stanley of whether two nonisomorphic trees can have the same Upolynomial (or, equivalently, the same chromatic symmetric function). We consider the Ukpolynomial, which is a restricted version of Upolynomial, and construct, for any given kk, nonisomorphic trees with the same Ukpolynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the Upolynomial up to isomorphism.

Threshold functions and Poisson convergence for systems of equations in random sets
http://hdl.handle.net/2117/111677
Threshold functions and Poisson convergence for systems of equations in random sets
Rué Perna, Juan José; Spiegel, Christoph; Zumalacárregui, Ana
We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sumfree sets, Bh[g]Bh[g]sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “AAcontains a nontrivial solution of M·x=0M·x=0” where AA is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1,…,n}{1,…,n}. Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the distribution of the number of nontrivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.
Juanjo Rué was partially supported by the FP7PEOPLE2013CIG Project CountGraph (ref. 630749), the Spanish MICINN Projects MTM201454745P and MTM201456350P, the DFG within the Research Training Group Methods for Discrete Structures (ref. GRK1408), and the Berlin Mathematical School. Christoph Spiegel was supported by a Berlin Mathematical School Scholarship. Ana Zumalacárregui is supported by the Australian Research Council Grant DP140100118. The first and third authors started this project while financed by the MTM201122851 Grant (Spain) and the ICMAT Severo Ochoa Project SEV20110087 (Spain).
Mon, 11 Dec 2017 11:59:37 GMT
http://hdl.handle.net/2117/111677
20171211T11:59:37Z
Rué Perna, Juan José
Spiegel, Christoph
Zumalacárregui, Ana
We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sumfree sets, Bh[g]Bh[g]sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “AAcontains a nontrivial solution of M·x=0M·x=0” where AA is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1,…,n}{1,…,n}. Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the distribution of the number of nontrivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.
Juanjo Rué was partially supported by the FP7PEOPLE2013CIG Project CountGraph (ref. 630749), the Spanish MICINN Projects MTM201454745P and MTM201456350P, the DFG within the Research Training Group Methods for Discrete Structures (ref. GRK1408), and the Berlin Mathematical School. Christoph Spiegel was supported by a Berlin Mathematical School Scholarship. Ana Zumalacárregui is supported by the Australian Research Council Grant DP140100118. The first and third authors started this project while financed by the MTM201122851 Grant (Spain) and the ICMAT Severo Ochoa Project SEV20110087 (Spain).

Corrigendum to"On the limiting distribution of the metric dimension for random forests" [European J. Combin. 49 (2015) 6889]
http://hdl.handle.net/2117/111651
Corrigendum to"On the limiting distribution of the metric dimension for random forests" [European J. Combin. 49 (2015) 6889]
Mitsche, Dieter; Rué Perna, Juan José
In the paper ”On the limiting distribution of the metric dimension for random forests” the metric dimension ß(G) of sparse G(n, p) with p = c/n and c < 1 was studied (Theorem 1.2). In the proof of this theorem, for the convergence in distribution Stein’s Method was applied incorrectly
Mon, 11 Dec 2017 10:25:46 GMT
http://hdl.handle.net/2117/111651
20171211T10:25:46Z
Mitsche, Dieter
Rué Perna, Juan José
In the paper ”On the limiting distribution of the metric dimension for random forests” the metric dimension ß(G) of sparse G(n, p) with p = c/n and c < 1 was studied (Theorem 1.2). In the proof of this theorem, for the convergence in distribution Stein’s Method was applied incorrectly

Counting configurationfree sets in groups
http://hdl.handle.net/2117/111650
Counting configurationfree sets in groups
Rué Perna, Juan José; Serra Albó, Oriol; Vena Cros, Lluís
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in nonabelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.
Mon, 11 Dec 2017 10:18:23 GMT
http://hdl.handle.net/2117/111650
20171211T10:18:23Z
Rué Perna, Juan José
Serra Albó, Oriol
Vena Cros, Lluís
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in nonabelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.

Isolated cycles of critical random graphs
http://hdl.handle.net/2117/111536
Isolated cycles of critical random graphs
Noy Serrano, Marcos; Rasendrahasina, Vonjy; Ravelomanana, Vlady; Rué Perna, Juan José
Consider the ErdosRenyi random graph G(n, M) built with n vertices and
M edges uniformly randomly chosen from the set of n2 edges. Let L be a set of positive integers. For any number of edges M 6n/2 + O(n 2/3 ), we compute – via analytic combinatorics – the number of isolated cycles of G(n, M) whose length is in L.
Mon, 04 Dec 2017 12:51:16 GMT
http://hdl.handle.net/2117/111536
20171204T12:51:16Z
Noy Serrano, Marcos
Rasendrahasina, Vonjy
Ravelomanana, Vlady
Rué Perna, Juan José
Consider the ErdosRenyi random graph G(n, M) built with n vertices and
M edges uniformly randomly chosen from the set of n2 edges. Let L be a set of positive integers. For any number of edges M 6n/2 + O(n 2/3 ), we compute – via analytic combinatorics – the number of isolated cycles of G(n, M) whose length is in L.

Random strategies are nearly optimal for generalized van der Waerden Games
http://hdl.handle.net/2117/111534
Random strategies are nearly optimal for generalized van der Waerden Games
Kusch, C.; Rué Perna, Juan José; Spiegel, Christoph; Szabó, T.
In a (1 : q) MakerBreaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], who studied biased Hgames, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [Beck, J., Van der Waerden and Ramsey type games, Combinatorica, 1 (1981), 103–116] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], the random strategy for Maker is again the best known strategy.
Mon, 04 Dec 2017 12:43:19 GMT
http://hdl.handle.net/2117/111534
20171204T12:43:19Z
Kusch, C.
Rué Perna, Juan José
Spiegel, Christoph
Szabó, T.
In a (1 : q) MakerBreaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], who studied biased Hgames, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [Beck, J., Van der Waerden and Ramsey type games, Combinatorica, 1 (1981), 103–116] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], the random strategy for Maker is again the best known strategy.