Wilfried Imrich

On optimal embeddings of metrics in graphs

Abstract This paper extends previous results of the authors. In particular, non-treerealizable metrics are investigated and it is shown that every finite metric has an optimal realization by a graph.

Über das schwache kartesische Produkt von Graphen

Abstract It is shown that every connected graph has a unique prime factor decomposition with respect to the weak Cartesian product. The resulting close relationship between the automorphism group of a connected graph and the automorphism groups of its prime factors is used to derive theorems about the transitivity, regularity, and primitivity of these groups. With minor modifications all results also hold for set systems.

On groups of polyhedral graphs

It is shown that every group of sense preserving automorphisms of a polyhedral graph is isomorphic to a group of rotations of the sphere.

Zehnpunktige kubische Graphen

An der Rechenanlage der Technischen Hochschule Wien wurden im Jahre 1966 von Gerd Baron alle zusammenhangenden kubischen Graphen mit zehn Knoten bestimmt. Dabei erhob sich die Frage, ob man diese Graphen nicht auch ohne Komputer auf einfache Art bestimmen konne. Die Struktur dieser 19 verschiedenen Graphen legte eine Klasseneinteilung nach der Zahl und Lage der auftretenden Dreiecke nahe. Diese Einteilung fuhrt auch tatsachlich zum Ziel, wie in dieser Arbeit gezeigt werden soll.

On products of graphs and regular groups

A graphX is called a graphical regular representation (GRR) of a groupG if the automorphism group ofX is regular and isomorphic toG. Watkins and Nowitz have shown that the direct productG×H of two finite groupsG andH has aGRR if both factors have aGRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complementX′ of a finite or infinite graphX is prime with respect to cartesian multiplication ifX is composite and not one of six exceptional graphs.

On the maximal distance of spanning trees

Using Ores definition of the distance of spanning trees in a connected graph G, we determine the maximal distance a spanning tree may have from a given spanning tree and develop an algorithm for the construction of two spanning trees with maximal distance. It is also shown that the maximal distance of spanning tress in G is equal to the cyclomatic number c(G) of G, if G has no bridges and if c(G)≤min(5, |G|−1).