MD - Matemàtica Discreta
http://hdl.handle.net/2117/3546
2017-05-23T03:37:09ZA lower bound for the size of a Minkowski sum of dilates
http://hdl.handle.net/2117/104404
A lower bound for the size of a Minkowski sum of dilates
Hamidoune, Yayha Old; Rué Perna, Juan José
Let A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |A + k·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author.
Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2·A + k·A|=(k+2)|A|-k2-k+2. Notice that |2·P+k·P|=(k+2)|P|-2k, if P is an arithmetic progression.
2017-05-15T09:11:57ZHamidoune, Yayha OldRué Perna, Juan JoséLet A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |A + k·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author.
Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2·A + k·A|=(k+2)|A|-k2-k+2. Notice that |2·P+k·P|=(k+2)|P|-2k, if P is an arithmetic progression.Asymptotic enumeration of non-crossing partitions on surfaces
http://hdl.handle.net/2117/104401
Asymptotic enumeration of non-crossing partitions on surfaces
Rué Perna, Juan José; Sau, Ignasi; Thilikos Touloupas, Dimitrios
We generalize the notion of non-crossing partition on a disk to general surfaces
with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of S
2017-05-15T08:44:00ZRué Perna, Juan JoséSau, IgnasiThilikos Touloupas, DimitriosWe generalize the notion of non-crossing partition on a disk to general surfaces
with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SOuterplanar obstructions for the vertex feedback set
http://hdl.handle.net/2117/104395
Outerplanar obstructions for the vertex feedback set
Rué Perna, Juan José; Thilikos Touloupas, Dimitrios; Stavropoulos, Konstantinos
For k = 1, let Fk be the class of graphs that contains k vertices meeting all its cycles. The minor-obstruction set for Fk is the set obs(Fk) containing all minor-minimal graphs that do not belong to Fk. We denote by Yk the set of all outerplanar graphs in obs(Fk). I
2017-05-15T07:47:43ZRué Perna, Juan JoséThilikos Touloupas, DimitriosStavropoulos, KonstantinosFor k = 1, let Fk be the class of graphs that contains k vertices meeting all its cycles. The minor-obstruction set for Fk is the set obs(Fk) containing all minor-minimal graphs that do not belong to Fk. We denote by Yk the set of all outerplanar graphs in obs(Fk). IOn the error term of the logarithm of the lcm of quadratic sequences
http://hdl.handle.net/2117/104311
On the error term of the logarithm of the lcm of quadratic sequences
Rué Perna, Juan José; Sarka, Paulius; Zumalacárregui, Ana
We study the logarithm of the least common multiple of the sequence of integers given by 12 + 1, 2 2 + 1, . . . , n2 + 1. Using a result of Homma [4] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo
2017-05-11T10:25:33ZRué Perna, Juan JoséSarka, PauliusZumalacárregui, AnaWe study the logarithm of the least common multiple of the sequence of integers given by 12 + 1, 2 2 + 1, . . . , n2 + 1. Using a result of Homma [4] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by CillerueloOn the number of nonzero digits of some integer sequences
http://hdl.handle.net/2117/104309
On the number of nonzero digits of some integer sequences
Cilleruelo, Javier; Luca, Florián; Rué Perna, Juan José; Zumalacárregui, Ana
Let b = 2 be a fixed positive integer. We show for a wide variety
of sequences {an}8n=1 that for most n the sum of the digits of an in base b is at least cb log n, where cb is a constant depending on b and on the sequence
2017-05-11T09:55:21ZCilleruelo, JavierLuca, FloriánRué Perna, Juan JoséZumalacárregui, AnaLet b = 2 be a fixed positive integer. We show for a wide variety
of sequences {an}8n=1 that for most n the sum of the digits of an in base b is at least cb log n, where cb is a constant depending on b and on the sequenceAsymptotic study of subcritical graph classes
http://hdl.handle.net/2117/104308
Asymptotic study of subcritical graph classes
Drmota, Michael; Fusy, Éric; Kang, Mihyun; Kraus, Veronika; Rué Perna, Juan José
We present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works in both the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp., $g_n$) of labelled (resp., unlabelled) graphs on n vertices from a subcritical graph class ${\mathcal{G}}=\cup_n {\mathcal{G}_n}$ satisfies asymptotically the universal behavior $g_n = c \!n^{-5/2} \!\gamma^n \! (1+o(1))$ for computable constants $c,\gamma$, e.g., $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from $\mathcal{G}_n$ converges (after rescaling) to a normal law as $n\to\infty$.
2017-05-11T09:45:20ZDrmota, MichaelFusy, ÉricKang, MihyunKraus, VeronikaRué Perna, Juan JoséWe present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works in both the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp., $g_n$) of labelled (resp., unlabelled) graphs on n vertices from a subcritical graph class ${\mathcal{G}}=\cup_n {\mathcal{G}_n}$ satisfies asymptotically the universal behavior $g_n = c \!n^{-5/2} \!\gamma^n \! (1+o(1))$ for computable constants $c,\gamma$, e.g., $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from $\mathcal{G}_n$ converges (after rescaling) to a normal law as $n\to\infty$.Dynamic programming for graphs on surfaces
http://hdl.handle.net/2117/104306
Dynamic programming for graphs on surfaces
Rué Perna, Juan José; Sau, Ignasi; Thilikos, Dimitrios
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.
2017-05-11T09:30:07ZRué Perna, Juan JoséSau, IgnasiThilikos, DimitriosWe provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Random cubic planar graphs revisited
http://hdl.handle.net/2117/104259
Random cubic planar graphs revisited
Noy Serrano, Marcos; Requile, Clement; Rué Perna, Juan José
The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n vertices
2017-05-10T10:54:16ZNoy Serrano, MarcosRequile, ClementRué Perna, Juan JoséThe goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n verticesOn the fractional Parts of a^n/n
http://hdl.handle.net/2117/104257
On the fractional Parts of a^n/n
Cilleruelo, Javier; Luca, Florian; Kumchev, Angel; Rué Perna, Juan José; Shparlinski, Igor
We give various results about the distribution of the sequence {a
n/n}n=1 modulo 1, where a = 2 is a fixed integer. In particular, we find and infinite subsequence A such that {a n/n}n¿A is well distributed.
Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this sequence hit
every interval J ¿ [0, 1] of length |J | = cN -0.47
2017-05-10T10:40:22ZCilleruelo, JavierLuca, FlorianKumchev, AngelRué Perna, Juan JoséShparlinski, IgorWe give various results about the distribution of the sequence {a
n/n}n=1 modulo 1, where a = 2 is a fixed integer. In particular, we find and infinite subsequence A such that {a n/n}n¿A is well distributed.
Also we show that for any constant c > 0 and a sufficiently large N, the fractional parts of the first N terms of this sequence hit
every interval J ¿ [0, 1] of length |J | = cN -0.47Del teorema de los 4 colores a la gravedad cuántica: enumeración de mapas
http://hdl.handle.net/2117/104256
Del teorema de los 4 colores a la gravedad cuántica: enumeración de mapas
Rué Perna, Juan José
Es más que habitual que cuestiones elementales no tengan una respuesta sencilla.
Esto ocurre con el enigma con el que iniciaremos esta aventura hacia la combinatoria
de los mapas. Deseamos pintar los distintos términos municipales de nuestra província
de origen, de tal modo que dos regiones colindantes reciban un color distinto
2017-05-10T10:27:11ZRué Perna, Juan JoséEs más que habitual que cuestiones elementales no tengan una respuesta sencilla.
Esto ocurre con el enigma con el que iniciaremos esta aventura hacia la combinatoria
de los mapas. Deseamos pintar los distintos términos municipales de nuestra província
de origen, de tal modo que dos regiones colindantes reciban un color distinto