arXiv subjects
Publications and source records attributed to Levente Szemerédi.
We give almost sure convergence rate bounds of ratio consensus algorithms when the protocol can be reformulated to be linear updates of vector values on a possibly larger, augmented network. This is an improvement of the results of Gerencsér and Gerencsér from 2021 by allowing zero values on auxiliary nodes infinitely often which makes the technique applicable to a much larger family of algorithms.
In this paper, we present a new factor of IID process based on the local algorithm introduced by Díaz, Serna, and Wormald (2007). This new approach allows us to improve the previously known upper bounds on the minimum and maximum bisection width and the maximum cut of random d-regular graphs for d > 4 by introducing a new recoloring phase after the termination of the original algorithm. As an application, we show that random 5-regular graphs asymptotically almost surely admit an internal partition, i.e., a partition of the vertex set into two nonempty classes so that every vertex has at least half of its neighbors in its own class.
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.