130 research outputs found
Del teorema de los 4 colores a la gravedad cuántica: enumeración de mapas
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 tĂ©rminos municipales de nuestra provĂncia
de origen, de tal modo que dos regiones colindantes reciban un color distintoPostprint (published version
A lower bound for the size of a Minkowski sum of dilates
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.Postprint (author's final draft
On the error term of the logarithm of the lcm of quadratic sequences
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 CillerueloPostprint (updated version
Asymptotic enumeration of non-crossing partitions on surfaces
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 SPostprint (author's final draft
The rado multiplicity problem in vector spaces over finite fields
We study an analogue of the Ramsey multiplicity problem for additive structures, establishing the minimum number of monochromatic -APs in -colorings of and obtaining the first non-trivial lower bound for the minimum number of monochromatic -APs in -colorings of . The former parallels results by Cumings et al.~\cite{CummingsEtAl_2013} in extremal graph theory and the latter improves upon results of Saad and Wolf~\cite{SaadWolf_2017}. Lower bounds are notably obtained by extending the flag algebra calculus of Razborov~\cite{razborov2007flag}.Peer ReviewedPostprint (author's final draft
On a problem of Sárközy and Sós for multivariate linear forms
We prove that for pairwise co-prime numbers k1,...,kd = 2 there does not exist
any infinite set of positive integers A such that the representation function rA(n) =
#{(a1,...,ad) Âż Ad : k1a1 + ... + kdad = n} becomes constant for n large enough.
This result is a particular case of our main theorem, which poses a further step
towards answering a question of S´ark¨ozy and S´os and widely extends a previous
result of Cilleruelo and Ru´e for bivariate linear forms (Bull. of the London Math.
Society 2009).Postprint (author's final draft
Domino tilings of the Aztec Diamond
Imagine you have a cutout from a piece of squared
paper and a pile of dominoes, each of which can cover
exactly two squares of the squared paper. How many
different ways are there to cover the entire paper
cutout with dominoes? One specific paper cutout can
be mathematically described as the so-called Aztec
Diamond, and a way to cover it with dominoes is
a domino tiling. In this snapshot we revisit some
of the seminal combinatorial ideas used to enumerate
the number of domino tilings of the Aztec Diamond.
The existing connection with the study of the
so-called alternating-sign matrices is also explored.Peer ReviewedPostprint (published version
Subgraph statistics in subcritical graph classes
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.Postprint (author's final draft
Enumeration of labeled 4-regular planar graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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.Peer ReviewedPostprint (author's final draft
Random cubic planar graphs revisited
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 verticesPostprint (author's final draft
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