Articles de revista http://hdl.handle.net/2117/3547 2020-02-22T23:37:09Z 2020-02-22T23:37:09Z On the number of labeled graphs of bounded treewidth Baste, Julien Noy Serrano, Marcos Sau, Ignasi http://hdl.handle.net/2117/115956 2020-02-12T22:41:23Z 2018-04-04T14:01:07Z 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 well-known formula for the number of labeled lambda-trees, 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 proper-pathwidth at most k . 2018-04-04T14:01:07Z 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 well-known formula for the number of labeled lambda-trees, 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 proper-pathwidth at most k . Transformation and decomposition of clutters into matroids Martí Farré, Jaume Mier Vinué, Anna de http://hdl.handle.net/2117/112728 2020-02-12T22:01:13Z 2018-01-12T13:52:45Z 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 non-empty 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. 2018-01-12T13:52:45Z 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 non-empty 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 U-polynomial and the Prouhet–Tarry–Escott problem Aliste Prieto, José Mier Vinué, Anna de Zamora, José http://hdl.handle.net/2117/112678 2020-02-12T22:01:11Z 2018-01-11T14:04:18Z On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem Aliste Prieto, José; Mier Vinué, Anna de; Zamora, José This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given kk, non-isomorphic trees with the same Uk-polynomial. 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 U-polynomial up to isomorphism. 2018-01-11T14:04:18Z Aliste Prieto, José Mier Vinué, Anna de Zamora, José This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given kk, non-isomorphic trees with the same Uk-polynomial. 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 U-polynomial up to isomorphism. Threshold functions and Poisson convergence for systems of equations in random sets Rué Perna, Juan José Spiegel, Christoph Zumalacárregui, Ana http://hdl.handle.net/2117/111677 2020-02-12T22:47:03Z 2017-12-11T11:59:37Z 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, sum-free sets, Bh[g]Bh[g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “AAcontains a non-trivial 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 non-trivial 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 FP7-PEOPLE-2013-CIG Project CountGraph (ref. 630749), the Spanish MICINN Projects MTM2014-54745-P and MTM2014-56350-P, 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 MTM2011-22851 Grant (Spain) and the ICMAT Severo Ochoa Project SEV-2011-0087 (Spain). 2017-12-11T11:59:37Z 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, sum-free sets, Bh[g]Bh[g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “AAcontains a non-trivial 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 non-trivial 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 FP7-PEOPLE-2013-CIG Project CountGraph (ref. 630749), the Spanish MICINN Projects MTM2014-54745-P and MTM2014-56350-P, 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 MTM2011-22851 Grant (Spain) and the ICMAT Severo Ochoa Project SEV-2011-0087 (Spain). Corrigendum to"On the limiting distribution of the metric dimension for random forests" [European J. Combin. 49 (2015) 68-89] Mitsche, Dieter Rué Perna, Juan José http://hdl.handle.net/2117/111651 2020-02-12T22:47:03Z 2017-12-11T10:25:46Z Corrigendum to"On the limiting distribution of the metric dimension for random forests" [European J. Combin. 49 (2015) 68-89] 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 2017-12-11T10:25:46Z 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 configuration-free sets in groups Rué Perna, Juan José Serra Albó, Oriol Vena Cros, Lluís http://hdl.handle.net/2117/111650 2020-02-12T22:47:03Z 2017-12-11T10:18:23Z Counting configuration-free 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 non-abelian 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. 2017-12-11T10:18:23Z 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 non-abelian 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. Random strategies are nearly optimal for generalized van der Waerden Games Kusch, C. Rué Perna, Juan José Spiegel, Christoph Szabó, T. http://hdl.handle.net/2117/111534 2020-02-12T22:00:47Z 2017-12-04T12:43:19Z 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) Maker-Breaker 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 H-games, 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. 2017-12-04T12:43:19Z Kusch, C. Rué Perna, Juan José Spiegel, Christoph Szabó, T. In a (1 : q) Maker-Breaker 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 H-games, 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. Enumeration of labeled 4-regular planar graphs Noy Serrano, Marcos Requile, Clement Rué Perna, Juan José http://hdl.handle.net/2117/109823 2020-02-12T22:00:14Z 2017-11-06T09:02:10Z Enumeration of labeled 4-regular planar graphs Noy Serrano, Marcos; Requile, Clement; Rué Perna, Juan José In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps. © <year>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ 2017-11-06T09:02:10Z Noy Serrano, Marcos Requile, Clement Rué Perna, Juan José In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps. Counting outerplanar maps Geffner, I. Noy Serrano, Marcos http://hdl.handle.net/2117/108706 2020-02-12T21:29:55Z 2017-10-16T08:23:36Z Counting outerplanar maps Geffner, I.; Noy Serrano, Marcos A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an e cient scheme for encoding simple outerplanar maps. 2017-10-16T08:23:36Z Geffner, I. Noy Serrano, Marcos A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an e cient scheme for encoding simple outerplanar maps. Subgraph statistics in subcritical graph classes Drmota, Michael Ramos Garrido, Lander Rué Perna, Juan José http://hdl.handle.net/2117/108598 2020-02-12T21:59:33Z 2017-10-10T13:38:22Z Subgraph statistics in subcritical graph classes Drmota, Michael; Ramos Garrido, Lander; Rué Perna, Juan José Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs. 2017-10-10T13:38:22Z Drmota, Michael Ramos Garrido, Lander Rué Perna, Juan José Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.