Project APVV-51-009605

List of researchers participating in the project

Project objectives

 Detailed description of objectives

Many problems of classical mathematics can be reduced to problems of Discrete mathematics. On the other hand, the development of Informatics, Information technologies, Statistics and the whole range of directions in Natural sciences gives strong motivations for solving some concrete tasks of Discrete mathematics. In our project we shall deal mainly with the following topics:

We aim at deriving and proving classification theorems for some families of large discrete groups of automorphisms of surfaces; large here means that the order is at least o(g), where o(g) is a function of genus.

Classification of regular embeddings of graphs. A 2-cell embedding of a graph into a surface is called regular if its automorphism group acts regularly on arcs. We consider the classification problem for several popular families of graphs such as complete bipartite graphs or n-dimensional cubes, for instance.

Chirality phenomenon. Maps, hypermaps and other related structures may appear in two non-isomorphic forms (enantiomers) , left- and right-oriented. We want to investigate the chirality phenomenon in a detail with the aim to derive chirality measures. We shall concetrate our investigation on regular maps and hypermaps and on fallerines.

Map enumeration and cyclic groups of automorphisms of surfaces. Our aim is to derive an enumeration formula for the number of oriented maps of given genus and given number of edges. We shall consider a similar problem in the category of unoriented maps. In both cases cyclic groups of automorphisms of surfaces play an important role.

Vertex-transitive maps of small genera. Our aim is to extend known classification of regular maps onto a classification of vertex-transitive maps.

Classification of compact connected 3-manifolds is not known. We shall concetrate on combinatorial and algorithmic aspects of the problem with the aim to create catalogues of small 3-manifolds.

The main goal of our research is to study the layouts of graphs and their properties. We will concentrate ourselves on specific classes of graphs which represent communication networks, which are of main interest in computer science.

Full but not strong dualities at the finite level. The aim is to find more examples of full but not strong dualities at the finite level for different quasi-varieties of algebras and a principle how to construct such dualities.

The pathology of full dualities. The aim is to explore the pathology of full dualities and to understand their behaviour for total structures.

The relationship between full and strong dualities for total structures. We aim at understanding this relationship for structures with total operations and relations but with no partial operations.

Full Duality Compactness Theorem. We plan to explore which versions of Duality Compactness Theorem for dualities, full dualities and strong dualities are valid.

Decompositions of snarks. Investigation of structure of snarks is closely related to some deep problems in discrete mathematics such as The Four Colour Theorem, 4- and 5-flow conjectures and The Cycle Double Cover Conjecture. We shall try to find a formulation of Six-decomposition Theorem for snarks and prove it.