Showing papers by "Michael Drmota published in 2014"

The maximum degree of random planar graphs

Abstract: Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1 − o(1) asymptotically equal to c log n, where c a 2.529 is determined explicitly. A similar result is also true for random 2-connected planar graphs.Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.

Subsequences of automatic sequences and uniform distribution.

Abstract: Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they cannot be used as quasi-random sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares. This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this article we survey these recent developments, comment on the proof methods and formulate quite general conjectures. We also present a new result on the subsequence along primes of so-called invertible automatic sequences.

Disproof of a conjecture by Rademacher on partial fractions

Abstract: In his book Topics in Analytic Number Theory (1973), Hans Rademacher considered the generating function of integer partitions into at most N parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19 (2013)), who gave strong numerical evidence against the conjecture.

Almost every tree with m edges decomposes K2m,2m

Abstract: We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K2m,2m, a result connected to a conjecture of Graham and Haggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size. © Cambridge University Press 2013.

Limit Laws for the Number of Groups formed by Social Animals under the Extra Clustering Model

Abstract: We provide a complete description of the limiting behaviour of the number Xn of groups that are formed by social animals when the number n of animals tends to infinity. The analysis is based on a random model by Durand, Blum and Francois, where it is assumed that groups are formed more likely by animals which are genetically related. The random variable Xn can be described by a stochastic recurrence equation that is very similar to equations that occur in the stochastic analysis of divide-and-conquer algorithms although it does not fall into already known cases. In particular, we obtain (in the most interesting) "neutral model" a curious central limit theorem, where the normalizing factor is p Var(Xn)=2. In the non-neutral (or extra clustering) cases the results are completely different. We obtain

A Gaussian limit process for optimal FIND algorithms

Abstract: We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on cadlag functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.