# Lecture by Balázs Patkós "Turán problems with dergee conditions"

Balázs Patkós will visit MIPT on December 18 2019. He will give a lecture on "Turán problems with dergee conditions" that will take place in Cifra 2.35 auditorium. The lecture is a part of "Seminar on Discrete Mathematics" and is organazed by our Laboratory and Phystech School of Applied Mathematics and Informatics.

December 18 | 17:00 | Cifra 2.35 |

**Abstract:**

Turán problems ask for the maximum number $ex(n,F)$ of edges that an $n$-vertex graph $H$ can have without containing a copy of the forbidden graph $F$. These problems are the starting points of extremal graph theory and there have been an enormous amount of research in the area in the past century. There exist many generalizations and variants to this kind of problems. In my talk, I will survey some recently introduced notions and the first couple of results concerning these notions all of which involve the degrees of either all vertices of the graph $H$ or of all vertices of the copy of $F$ in $H$. More precisely, I will talk about the following problems.

A subgraph $H$ of $G$ is *singular* if the vertices of $H$ either have the same degree in $G$ or have pairwise distinct degrees in $G$. The largest number of edges of a graph on $n$ vertices that does not contain a singular copy of $H$ is denoted by $T_S(n,H)$. We also explore the connection to the so-called $H$-WORM colorings (colorings without rainbow or monochromatic copies of $H$) and obtain new results regarding the largest number of edges that a graph with an $H$-WORM coloring can have.

The *regular Turán number* $rex(n,F)$ is the maximum number of edges that an $n$-vertex *regular* graph $H$ can have without containing a copy of $F$.

Everyone is invited to attend. The language of the lecture is English. The event is aimed at master and graduate students, as well as researchers in the field of combinatorics.