Horst Sachs

Perfect matchings in hexagonal systems

A simple algorithm is developed which allows to decide whether or not a given hexagonal system has a perfect matching (and to find such a matching). This decision is also of chemical relevance since a hexagonal system is the skeleton of a benzenoid hydrocarbon molecule if and only if it has a perfect matching.

Coin graphs, polyhedra, and conformal mapping

Abstract This is a brief report on some interesting theorems and their interconnections. All of these results are known, some of them have only recently been proved. The key result, however, though found already in 1935, was almost forgotten and its original proof remained unnoticed until very recently. Therefore, it seems worthwhile to the author to take the opportunity to inform the graph theoretic community about some facts which appear to be not so well known.

Spectra of toroidal graphs

An n-fold periodic locally finite graph in the Euclidean n-space may be considered the parent of an infinite class of n-dimensional toroidal finite graphs. An elementary method is developed that allows the characteristic polynomials of these graphs to be factored, in a uniform manner, into smaller polynomials, all of the same size. Applied to the hexagonal tessellation of the plane (the graphite sheet), this method enables the spectra and corresponding orthonormal eigenvector systems for all toroidal fullerenes and (3, 6)-cages to be explicitly calculated. In particular, a conjecture of P.W. Fowler on the spectra of (3, 6)-cages is proved.

On constructive methods in the theory of colour-critical graphs

Abstract Some 30 years ago, G.A. Dirac, T. Gallai and G. Hajos founded and developed the theory of colour-critical graphs as an important method in graph colouring theory. Since then, about 65 papers have been written on this subject containing many ideas how to construct colour-critical graphs with some specified properties. The authors survey the use of constructive methods and, tentatively, discuss their power as well as their limitations.

On colour critical graphs

A brief survey of the theory of colour critical graphs, with an emphasis on constructive methods, is given.

Calculating the numbers of perfect matchings and of spanning trees, Pauling's orders, the characteristic polynomial, and the eigenvectors of a benzenoid system

A survey of relevant papers is given, and five simple and simply handleable algorithms of low complexity based on results contained in these papers are described (without proofs):

Monotone path systems in simple regions

A monotone path system (MPS) is a finite set of pairwise disjoint paths (polygonal areas) in thexy-plane such that every horizontal line intersects each of the paths in at most one point. A MPS naturally determines a “pairing” of its top points with its bottom points. We consider a simple polygon Δ in thexy-plane wich bounds the simple polygonal (closed) regionD. LetT andB be two finite, disjoint, equicardinal sets of points ofD. We give a good characterization for the existence of a MPS inD which pairsT withB, and a good algorithm for finding such a MPS, and we solve the problem of finding all MPSs inD which pairT withB. We also give sufficient conditions for any such pairing to be the same.

Contributions to a Characterization of the Structure of Perfect Graphs

Characterizing the structure of perfect graphs is one of the important, most actual, and most challenging unsolved problems of graph theory. The strong version of Claude Berges Perfect Graph Conjecture has given rise to numerous investigations of this problem. In this paper, results found by E. Olaru since 1969 are summarized.

Construction of Colour-Critical Graphs With Given Major-Vertex Subgraph

T. Gallai (1963) characterized the class L k of all subgraphs spanned by the low vertices for the class of k -critical graphs ( k ≥ 4) having ‘low’ (minor) vertices (i.e., vertices of valency k - 1). In this paper, the class H k of all subgraphs spanned by the ‘high’ (major) vertices (i.e., vertices of valencies ≥ k ) is characterized. A general principle is given which, for given H ɛ H k , enables arbitrarily many k -critical graphs having H as their high-vertex subgraph to be constructed. The main results are contained in Theorems 3.1 and 7.1.

Calculating the characteristic polynomial, eigenvectors and number of spanning trees of a hexagonal system

Three simple algorithms of low complexity are described (without proof): 1) Algorithm A enables the characteristic polynomial and the eigenspaces (eigenvectors, HMOs) to be calculated for all hexagonal systems (especially those representing benzenoid hydrocarbons). 2) Algorithm B enables the same to be done, in a more efficient way, for those hexagonal systems whose inner dual is a tree (representing catacondensed benzenoid hydrocarbons). 3) Algorithm C enables, in a particularly efficient way, the number of spanning trees of any hexagonal system to be determined