Michał Karoński

Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

A weighting of the edges of a graph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph. In this paper we show that such a weighting is possible from the weight set {1,2,3,4,5} for all graphs not containing components with exactly 2 vertices.

On Random Intersection Graphs: The Subgraph Problem

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

A central limit theorem for decomposable random variables with applications to random graphs

Abstract The application of Steins method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory. Problems which exhibit a dissociated structure and problems which do not are considered. Results are obtained for the number of copies of a given graph G in K ( n , p ), for the number of induced copies of G , for the number of isolated trees of order k ≥ 2, for the number of vertices of degree d ≥ 1, and for the number of isolated vertices.

A review of random graphs

This paper presents a review of results dealing with structural properties and numerical characteristics of random edge-subgraphs of complete graphs, with the connectedness of random graphs, as well as with random trees and forests.

A New Upper Bound for the Irregularity Strength of Graphs

A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices are all different. In this note we show that such a weighting is possible from the weight set {1,2,…,...

A faster distributed algorithm for computing maximal matchings deterministically

We show that maximal matchings can be computed deterministically in O(log4 n) rounds in the synchronous, message-passing model of computation. This improves on an earlier result by three log-factors.

The number of connected sparsely edged uniform hypergraphs

Abstract Certain families of d -uniform hypergraphs are counted. In particular, the number of connected d -uniform hypergraphs with r vertices and r + k hyperedges, where k = o(log r / log log r ), is found.

The Origins of the Theory of Random Graphs

The origins of the theory of random graphs are easy to pin down. Undoubtedly one should look at a sequence of eight papers co-authored by two great mathematicians: Paul Erdős and Alfred Renyi, published between 1959 and 1968:

Small cliques in random graphs

For d ≥ 1, a d-clique in a graph G is a complete d-vertex subgraph not contained in any larger complete subgraph of G. We investigate the limit distribution of the number of d-cliques in the binomial random graph G(n, p), p = p(n), n→∞.

Application of digital computers in NMR lanthanide shift analysis

Abstract For quantitative application of paramagnetic shifts observed in the NMR spectra of many organic molecules upon addition of rare earth chelates, the geometry of interaction between these molecules and the “shift reagent” must be known. The authors describe and evaluate a set of computer programs useful for routine analysis of data obtained by an NMR lanthanideinduced shift technique, namely for positioning of the lanthanide ion in reference to the substrate, and for automatic interpretation of the shifted spectra.