The Research School covers three topics in Graph Theory which are active fields of research and give the opportunity to present a wealth of open problems and existing techniques to attack them.
Many problems in Graph Theory can be formulated in terms of labelings of graphs, or weights given to vertices and edges with prescribed algebraic or arithmetic properties. Different types of labelings, structural properties associated to them and existencial and constructive techniques will be presented in the course, together with striking open conjectures in the area. Applications of these labelings to technology and computer science will be also covered. The closely related area of Graph decompositions of graphs into parts with particular properties will also be covered in the School, including packing and covering problems, which again collect a number of well-known open conjectures. The third topic covered in the School is related to cycles and paths in graphs, particularly Hamiltonian cycles, a good example of a hard computational problem with a large number of results. Closure techniques, structural conditions and forbidden subgraph characterizations will be covered in the course. Applications will be presented, including discussions on the Travelling Salesmen Problem.
The main purpose of the School is to have the participants make a start to attempt to solve some selected problems. This will be done in small groups under the guidance of the lecturers. The School will conclude with presentations of participants on their work.