## Research Grants

### Asymptotic properties of graphs on a surface

Since the foundation of the theory of random graphs by Erdős and Rényi five decades ago, various random graphs have been introduced and studied. One example is random graphs on a surface, in particular random planar graphs. Graphs on a 2-dimensional surface and related objects (e.g. planar graphs, triangulations) have been among the most studied objects in graph theory, enumerative combinatorics, discrete probability theory, and statistical physics. The main objectives of this project are to study the asymptotic properties and limit behaviour of random graphs on a surface (e.g. evolution, phase transition, critical behaviour, component size distribution) and to investigate enumerative and algorithmic aspects of unlabelled graphs on a surface (e.g. connectivity, symmetry, decomposition, random generation).

- supported by Austrian Science Fund (FWF), Grant no. P27290, 01.06.2015-31.05.2018
- Institute's members in this project: Mihyun Kang (PI), Philipp Sprüssel (Co-PI)

### Phase transitions and critical phenomena in random graphs

Random graph models that this project focuses on are random graph processes and random hypergraphs. The constraints imposed on these random graph models (in particular random graph processes) lead to difficulties in the analysis of their asymptotic behaviour, due to the long-term and/or global dependence between edges. To overcome these difficulties, new approaches have to be found. The main objective of this project is to advance analytic and probabilistic approaches and to apply them to analyse asymptotic behaviour of such complex random graph models. The scientific program of this project consists of two main themes, which are closely related in that both themes deal with phase transitions and critical phenomena.

- supported by Austrian Science Fund (FWF), Grant no. P26826, 01.05.2014-30.04.2017
- Institute's members in this project: Mihyun Kang (Principal Investigator), Tamas Makai (Postdoc), Christoph Koch (Doctoral Student)

### Doctoral Program Discrete Mathematics

The institute of Optimization and Discrete Mathematics takes part in the Doctoral Program "Discrete Mathematics", which offers an advanced PhD training and research program.

- supported by Austrian Science Fund (FWF), Grant no. W1230 I (2010-2014), W1230 II (2015-2019)
- Institute's members in this project:
- Mihyun Kang: Principal Investigator of Project 14. Random graphs on a surface
- Bettina Klinz: Principal Investigator of Project 7. Combinatorial optimisation problems

- One PhD position in Random Graphs and Combinatorics is available (deadline: 15 May 2015)

## Research Grants -- Completed

### Phase Transitions in Random Graphs and Random Graph Processes

The objectives of this project are to study the phase transitions in random graphs and random graph processes with constraints such as degree distribution, forbidden substructures, genus. The phase transition is a phenomenon observed in many fundamental problems from statistical physics, mathematics and theoretical computer science, including Potts models, graph colourings and satisfiability problem. The phase transition observed in the plethora of different random graph models refers to a phenomenon that there is a critical value of edge density such that adding a small number of edges around the critical value results in a dramatic change in the size of the largest components. It is our aim to further develop and apply new analytic approaches combined with counting and probabilistic methods, e.g. singularity analysis, differential equations method, to the study of the phase transitions in random graphs and random graph processes.

- supported by German Research Foundation (DFG), Grant no. KA 2748/3-1, 01.10.2011-30.11.2014
- Institute's members in this project: Mihyun Kang (Principal Investigator), Tamas Makai (Postdoc)