F50-02: Shape Characteristics of Planar Maps and Planar Graphs
This project part deals with the asymptotic analysis of shape parameters (like component structure, degree distribution of vertices, diameter) of random planar maps and graphs and related combinatorial structures. The enumeration of rooted maps is a classical subject, initiated by Tutte in the 1960's whereas the study of random planar graphs is a recent one. Since then many shape characteristics of random planar maps and graphs have been studied (degree distribution, maximum degree, core size, etc.). Recently the interest in planar maps and graphs has considerably increased. This is due to fundamental findings by Schaeffer (bijections for planar maps) and Gimenez and Noy (generating function approach to planar graphs). The PI (and his coauthors) has contributed to this development, too, in particular on the degree distribution and the maximum degree of several classes of random planar graphs.
The main goals of this project part are to settle important open questions in this field (like the asymptotics of unlabeled planar graphs, which is certainly the most challenging problem) and to develop a more systematic approach that works on a wide class of random graph classes. All parts follow the philosophy of it Analytic Combinatorics, that is, the combinatorial structure is translated into relations of generating functions, that are analyzed then with the help of complex analytic methods.