### Project 15: "Random graphs on a surface"

*This project is running since 2015.*

Principal investigator: Mihyun Kang

Graz University of Technology, Austria.

*Mentor for*: Kloas, Moosmüller; Carl.

### DK Student

**Second phase of the doctoral program:**

**Nicola del Guidice**(Italy; since September 2016)

*Mentors*: Michael Kerber, Christopher Dowden, Philipp Sprüssel.

### Associated Students

**Second phase of the doctoral program:****Christoph Koch**(Germany; April 2012 - November 2016)

*Email*: ckoch@math.tugraz.at

*Mentors*: Wolfgang Woess, Oliver Cooley.

*PhD Defense*: November 25, 2016.

*Referees*: Mihyun Kang (TU Graz), Michael Krivelevich (Tel Aviv University), and Angelika Steger (ETH Zürich).

*Examiners*: Angelika Steger, Wolfgang Woess.**Michael Moßhammer**(Austria; since October 2015)

*Email*: mosshammer@math.tugraz.at

*Mentors*: Johannes Wallner, Philipp Sprüssel.

*PhD Defense*: May 4, 2018.

*Referees*: M. Drmota (Vienna), M. Kang, W. Woess.

*Examiners*: M. Drmota (Vienna), M. Kang.

### Project description

The main research field of Mihyun Kang is probabilistic and enumerative combinatorics, random graphs and hypergraphs, planar graphs and graphs on surfaces, and randomised algorithms. The main objectives are to study asymptotic properties and limit behaviour (e.g. evolution, phase transition, critical behaviour, component size distribution) and to investigate structural, enumerative and algorithmic aspects (e.g. symmetry, decomposition, asymptotic number, random sampling). In comparison with the classical Erdos-Renyi random graphs, additional constraints imposed on random graphs (e.g. planarity, genus, degree) lead to serious difficulties in the analysis. To circumvent these difficulties and to achieve the objectives, problems are approached by means of the combination of complementary methods, such as probabilistic methods, graph theoretic methods, differential equations method, methods from analytic combinatorics (e.g. singularity analysis, saddle point method), and algorithmic methods (e.g. Boltzmann sampler).