Discrete mathematics (Finite Geometry and Graph Theory)

Finite geometry deals with geometric structures consisting of finitely many objects. Most of them can be described by the notion of a block-design consisting of a finite set of points and a collection of subsets of the point-set called blocks. The relevant geometric properties derive from the incidence relations which are imposed on the structure under consideration. Finite geometry includes the study of finite projective and affine spaces, in particular projective and affine planes (whose non-desarguesian models are far from being classified), and the so called circle-geometries (Möbius, Laguerre and Minkowski planes) that generalize the geometry of quadrics in 3-space. In recent years increasing interest has been devoted to the notion of a graph-design, that is a block-design in which each block has the "shape" of an assigned graph.
Structural properties of specified classes of graphs may well be of interest in their own right and so when adopting this point of view graphs become THE object to be investigated rather than forming a model for studying something else. So, for instance, the class of graphs admitting a Hamiltonian cycle is not yet fully understood. This area of discrete mathematics is famous for many problems admitting an elementary formulation but whose solution is not elementary at all: the 4-colour theorem is perhaps the most famous instance in this respect. As a matter of fact many conjectures in Graph Theory are still wide open and progress is slow. The study of large examples or putative counterexamples may well involve enumerative arguments and the use of computer techniques. Current research often involves graphs which are nowadays known as “snarks” (they are essentially cubic graphs not admitting a 3-edge-colouring).

[Ultimo aggiornamento: 02/02/2021 12:40:00]