MD  Matemàtica Discreta
http://hdl.handle.net/2117/3546
Mon, 22 Jan 2018 12:21:21 GMT
20180122T12:21:21Z

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.

Enumeration of labeled 4regular planar graphs
http://hdl.handle.net/2117/109823
Enumeration of labeled 4regular planar graphs
Noy Serrano, Marcos; Requile, Clement; Rué Perna, Juan José
In this extended abstract, we present the first combinatorial scheme for counting labeled 4regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4regular 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 byproduct, we can also compute the algebraic generating function counting labeled 3connected 4regular planar maps.
© <year>. This manuscript version is made available under the CCBYNCND 4.0 license http://creativecommons.org/licenses/byncnd/4.0/
Mon, 06 Nov 2017 09:02:10 GMT
http://hdl.handle.net/2117/109823
20171106T09:02:10Z
Noy Serrano, Marcos
Requile, Clement
Rué Perna, Juan José
In this extended abstract, we present the first combinatorial scheme for counting labeled 4regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4regular 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 byproduct, we can also compute the algebraic generating function counting labeled 3connected 4regular planar maps.

Counting outerplanar maps
http://hdl.handle.net/2117/108706
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.
Mon, 16 Oct 2017 08:23:36 GMT
http://hdl.handle.net/2117/108706
20171016T08:23:36Z
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
http://hdl.handle.net/2117/108598
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 seriesparallel graphs.
Tue, 10 Oct 2017 13:38:22 GMT
http://hdl.handle.net/2117/108598
20171010T13:38:22Z
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 seriesparallel graphs.