Series talks
Lecture series on random regular graphs
In this series we will get familiar with two tools needed to analyze random regular graphs: the configuration model (due to B. Bollobas) and the switching method (due to B. McKay). I invite you to choose one of the the four topics to present. After you read, I suggest meeting you and discussing the challenges and make suggestion what to omit (if it is hard to fit in a single talk) or how to expand (if things are going too easy).
Topics about dregular graphs, d fixed
 Limit distribution of number of cycles in R(n,d), d fixed (and the asymptotic number of dregular graphs that follows), section 9.2 in book 'Random graphs' by Janson and Luczak and Rucinski.
 Almost every dregular graph is dconnected (with room to spare), Thm 7.32 in book 'Random graphs' by Bollobas.
Topics about dregular graphs, d growing (the method of switchings)
 Asymptotic formula for number of graphs with given degree sequence, when max degree is o(n^{1/3}). Based on [MW90] pages 5259, can be simplified by assuming dregular graphs.
(Alternative source: Theorem 10.4 in the book 'Introduction to random graphs' by Frieze and Karonski).
 Generating uniformly a graph with given degree sequence, max degree o(n^{1/3}) in expected polynomial O(n^2k^4) time. Based on [MW90], pages 5962, you can assume that topic 3 is already presented; if you are doing well, you can present the better algorithm from Section 5); you can simplify the presentation by assuming dregular graphs.
If you are interested contact me at matas.sileikis at gmail.com
Introduction to Communication Complexity
Communication model seems to be good to prove unconditional lower bounds and still sufficiently strong to design some nontrivial algorithms. There are two players Alice and Bob in the standard communication model and they know some function f: X x Y > Z of two variables.
Alice gets x in X and Bob gets y in Y. Their goal is to communicate and then output f(x,y). Communication complexity of f is the minimal number of bits needed to send to compute the function f. In this series we introduce this model, show some basic technique and applications of lower bounds in the communication model.
The series is suitable for bachelor and master students. It is planned for 3 slots:
 Introduction of the communication model and its variants. Basic lower bounds technique.
 A little bit advanced techniques and applications.

 Information complexity and its applications  similar notion to communication complexity, however we measure sent information instead of bits (suitable for a master student).
 Multiparty communication model  models where we have several players instead of two.
For more information please write at koblich at iuuk.mff.cuni.cz.
Available individual papers
Minimum density of identifying codes of king grids by R. Dantas, R.M. Sampaio, and F. Havet Identifying code in a graph is a set of vertices such that every vertex of graph is uniquely defined by the set of adjacent vertices from the identifying code. Paper gives some bounds on minimum density of such codes using discharging methods. As usual for discharging methods, some proofs get rather technical and they should be omitted or only the main idea should be presented (this in particular applies to Theorems 8 and 9 in the paper).
Tiling the plane with equilateral triangles by Janos Pach, Gabor Tardos Let 𝒯 be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then 𝒯 is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar.
Acute Sets by Dmitriy Zakharov A set of points in R^d is acute if any three points from this set form an acute triangle. In this note we construct an acute set in R^d of size at least 1.618^d. We also present a simple example of an acute set of size at least 2^d/2.