Geometric graphs and applications

• Applicant: Professor Dr. Horst Martini
• Subject Area: Mathematics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 327566472

Geometric graphs are graphs whose vertices and edges are identified with points and (metric) line segments, respectively. Known examples are Delaunay triangulations or 1-skeletons of polytopes. As "geometrization" of an abstract incidence structure, the notion has many applications, in disciplines like optimization ("trees" in location science), discrete geometry (e.g., Erdös-type problems on metrically extremal point sets), convexity (for instance, polytope theory or bodies of constant width), and also in various non-Euclidean geometries (e.g., in general normed spaces). With this "Neuantrag", Prof. H. Martini (Applicant, TU Chemnitz/Germany) applies together with Prof. A. Kamal (Applicant, University teacher at Alquds University, Abu Dis Jerusalem/Palestine), and Prof. Yaakov S. Kupitz (Applicant, Lecturer at the Hebrew University in Jerusalem/Israel) for a trilateral research project referring to geometric graphs and their applications. We have chosen this subject since all of us work in this field already for many years and published jointly related papers. So we developed already a broad mixture of graph-theoretic and geometric methods which is promising in view of reaching really the goals of the described research program. Our already published results mainly refer to basic properties of geometric graphs as well as to applications of them. Now, within our project, we want to stay in such research directions, and within three years we want to write eight joint publications on the following topics. First, regarding fundamental properties of geometric graphs, we will investigate several classes of geometric graphs, which are extremal regarding various properties, where also colorings will play a role. Second, in applied direction we will study finite point sets with extremal diameter graphs (i.e., typical Erdös-type problems), problems in optimization (location science) and generalized partition problems for finite point sets. All these eight papers will be four-authors papers (with M. A. Perles as external coauthor). In the application we also describe the second joint three-years period (planned via "Fortsetzungsantrag"), which will be more of applied nature, with topics like: constructions of metrically extremal point sets in finite-dimensional real Banach spaces and further applications of geometric graphs in location science (i.e., optimization). Based on all this we hope that, with a deeper study of geometric graphs, our research project will forge links and create new or expand existing interactions between the mathematical fields of discrete and computational geometry, combinatorics and graph theory, convexity, as well as Minkowski geometry. Furthermore, the planned project will positively influence the work of our research groups (for example, lectures in research seminars when visiting each other, and refereeing activities regarding PhD students, with respect to all three universities).

• DFG Programme: Research Grants
• International Connection: Israel, Palestine
• International Co-Applicants: Dr. Abdullah Kamal; Dr. Yaakov S. Kupitz