Disproof of a conjecture about independent branchings in k-connected directed graphs

For each k ≥ 3, we construct a finite directed strongly k-connected graph D containing a vertex t with the following property: For any k spanning t-branchings, B₁, …, B_k in D (i. e., each B_i is a spanning tree in D directed toward t), there exists a vertex x ≠ t of D such that the k, x, t-paths in B₁, …, B_k are not pairwise openly disjoint. This disproves a well-known conjecture of Frank. © 1995, John Wiley & Sons, Inc.     

Heterofullerenes: Structure and property predictions,possible uses and synthesis proposals

Summary Substituting carbon atoms of fullerenes by heteroatoms and vacancies will lead to new and yet unknown spherical-shaped molecules termed hereafter as heterofullerenes. The enormous structural diversity of these molecules is investigated and their structural, electronic and thermochemical properties are predicted using semiempirical computations. Computational results for complexes with ions lead to the hypothesis that these molecules behave like microscopic Faraday cages in which the electrons concentrate on the outer side of the sphere. It is predicted that some of these heterofullerenes are structurally and electronically similar to phthalocyanines and related molecules but offer many additional advantages. Potential uses such as adding heterofullerenes to fullerene materials, as superior starting materials for the fabrication of diamonds, as catalysts in hydrogenation reactions, as components of materials dominated until now by phthalocyanines, etc., are discussed. Simple synthetic routes to these compounds that are based on minor alternations of existing methods for fullerene production are proposed. On the basis of the thermochemical calculations, we believe that the most promising possibility consists of using metal cyanide/graphite composite target rods instead of pure graphite rods as in a conventional fullerene synthesis.     

The Apollonius problem in four-dimensional Laguerre planes

The number of circles of a four-dimensional locally compact Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.