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

*This project is running since 2015.*

Principal investigator: Mihyun Kang

Graz University of Technology, Austria.

*Mentor for*: Dornelas; Kloas, Moosmüller; Carl.

Associated scientist: Joshua Erde

Graz University of Technology, Austria.

*Mentor for*: Do; Schmid

### DK Student

**Third phase of the doctoral program:****Tuan Anh Do**(Vietnam; October 2019–April 2023)

*Email*: do@math.tugraz.at

*Mentors*: Oswin Aichholzer, Joshua Erde.

*Thesis Title*: "Structural properties of sparse random graph models".

*PhD Defense*: April 28, 2023.

*Referees*: Michael Krivelevich (Tel Aviv University), Mihyun Kang, Will Perkins (Georgia Tech).

*Examiners*: Michael Krivelevich (Tel Aviv University), Will Perkins (Georgia Tech) .**Michael Missethan**(Austria; October 2019–May 2022)

*Email*: missethan@math.tugraz.at

*Mentors*: Michael Kerber, Philipp Sprüssel.

*Thesis Title*: "Global and local properties of random planar graphs".

*PhD Defense*: May 6, 2022.

*Referees*: M. Drmota (TU Wien), M. Kang (TU Graz), K. Panagiotou (LMU München).

*Examiners*: M. Drmota (TU Wien), K. Panagiotou (LMU München).

**Second phase of the doctoral program:****Nicola Del Giudice**(Italy; September 2016–June 2020)

*Email*: delgiudice@math.tugraz.at

*Mentors*: Michael Kerber, Oliver Cooley, Philipp Sprüssel.

*Thesis Title*: "Random hypergraphs and random simplicial complexes".

*PhD Defense*: June 19, 2020.

*Referees*: T. Łuczak (Adam Mickiewicz University), T. Müller (Groningen University) M. Kang.

*Examiners*: T. Łuczak (Adam Mickiewicz University), T. Müller (Groningen University).

### Associated Students

**Third phase of the doctoral program:****Anna Geisler**(Germany; since March 2023)

*Email*: geisler@math.tugraz.at

*Mentors*: Michael Kerber, Christoper Frei.

**Dominik Schmid**(Austria; since May 2022)

*Email*: schmid@math.tugraz.at

*Mentors*: Michael Kerber, Joshua Erde.

**Julian Zalla**(Germany; March 2019–May 2022)

*Email*: zalla@math.tugraz.at

*Mentors*: Peter Grabner, Oliver Cooley.

*Thesis Title*: "High-order substructures in random hypergraphs".

*PhD Defense*: May 20, 2022.

*Referees*: J. Böttcher (LSE), M. Kang, Y. Person (TU Ilmenau).

*Examiners*: J. Böttcher (LSE), Y. Person (TU Ilmenau).**Second phase of the doctoral program:****Christoph Koch**(Germany; April 2012 - November 2016)

*Email*: ckoch@math.tugraz.at

*Mentors*: Wolfgang Woess, Oliver Cooley.

*Thesis Title*: "Phase transition phenomena in random graphs and hypergraphs".

*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; October 2013 - May 2018)

*Email*: mosshammer@math.tugraz.at

*Mentors*: Johannes Wallner, Philipp Sprüssel.

*Thesis Title*: "Phase transitions and structural properties of random graphs on surfaces".

*PhD Defense*: May 4, 2018.

*Referees*: M. Drmota (TU Wien), M. Kang, C. McDiarmid (University of Oxford).

*Examiners*: M. Drmota (TU Wien), 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).