## Archive for the ‘KAIST Discrete Math Seminar’ Category

### Zi-Xia Song (宋梓霞), Ramsey numbers of cycles under Gallai colorings

Monday, October 14th, 2019

IBS/KAIST Joint Discrete Math Seminar

Ramsey numbers of cycles under Gallai colorings
Zi-Xia Song (宋梓霞)
University of Central Florida
2019/10/15 Tue 4:30PM-5:30PM (Room B232, IBS)
For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4$ vertices. For odd cycles, Bondy and Erd\H{o}s in 1973 conjectured that for all $k\ge1$ and $n\ge2$, $R_k(C_{2n+1})=n\cdot 2^k+1$. Recently, this conjecture has been verified to be true for all fixed $k$ and all $n$ sufficiently large by Jenssen and Skokan; and false for all fixed $n$ and all $k$ sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of $R_k(C_{2n})$ in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all $k$ and all $n$ under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings. Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.

### Alexandr V. Kostochka, Reconstructing graphs from smaller subgraphs

Monday, October 14th, 2019

IBS/KAIST Joint Discrete Math Seminar

Reconstructing graphs from smaller subgraphs
Alexandr V. Kostochka
University of Illinois at Urbana-Champaign
2019/10/10 Thu 4:30PM-5:30PM
A graph or graph property is $\ell$-reconstructible if it is determined by the multiset of all subgraphs obtained by deleting $\ell$ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each $\ell\in\mathbb N$, there is an integer $n=n(\ell)$ such that every graph with at least $n$ vertices is $\ell$-reconstructible. We show that for each $n\ge7$ and every $n$-vertex graph $G$, the degree list and connectedness of $G$ are $3$-reconstructible, and the threshold $n\geq 7$ is sharp for both properties.‌ We also show that all $3$-regular graphs are $2$-reconstructible.

### Alexandr V. Kostochka, On Ramsey-type problems for paths and cycles in dense graphs

Monday, October 14th, 2019

KAIST MathSci / IBS DIMAG Joint Colloquium

On Ramsey-type problems for paths and cycles in dense graphs
Alexandr V. Kostochka
University of Illinois at Urbana-Champaign
2019/10/08 Tue 4:30PM-5:30PM
A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph $G$ arrows a graph $H$ if for any coloring of the edges of $G$ with two colors, there is a monochromatic copy of $H$. In these terms, the above puzzle claims that the complete $6$-vertex graph $K_6$ arrows the complete $3$-vertex graph $K_3$. We consider sufficient conditions on the dense host graphs $G$ to arrow long paths and even cycles. In particular, for large $n$ we describe all multipartite graphs that arrow paths and cycles with $2n$ edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such $n$. Also for large $n$ we find which minimum degree in a $(3n-1)$-vertex graph $G$ guarantees that $G$ arrows the $2n$-vertex path. This yields a more recent conjecture of Schelp. This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu.

### Casey Tompkins, Extremal problems for Berge hypergraphs

Monday, October 14th, 2019

IBS/KAIST Joint Discrete Math Seminar

Extremal problems for Berge hypergraphs
Casey Tompkins
IBS Discrete Mathematics Group
2019/10/01 Tue 4:30PM-5:30PM
Given a graph $G$, there are several natural hypergraph families one can define. Among the least restrictive is the family $BG$ of so-called Berge copies of the graph $G$. In this talk, we discuss Turán problems for families $BG$ in $r$-uniform hypergraphs for various graphs $G$. In particular, we are interested in general results in two settings: the case when $r$ is large and $G$ is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when $G$ is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.

### Cory Palmer, A survey of Turán-type subgraph counting problems

Monday, October 14th, 2019

IBS/KAIST Joint Discrete Math Seminar

A survey of Turán-type subgraph counting problems
Cory Palmer
University of Montana, Missoula, MT
2019/09/19 Tue 4:30PM-5:30PM
Let $F$ and $H$ be graphs. The subgraph counting function $\operatorname{ex}(n,H,F)$ is defined as the maximum possible number of subgraphs $H$ in an $n$-vertex $F$-free graph. This function is a direct generalization of the Turán function as $\operatorname{ex}(n,F)=\operatorname{ex}(n,K_2,F)$. The systematic study of $\operatorname{ex}(n,H,F)$ was initiated by Alon and Shikhelman in 2016 who generalized several classical results in extremal graph theory to the subgraph counting setting. Prior to their paper, a number of individual cases were investigated; a well-known example is the question to determine the maximum number of pentagons in a triangle-free graph. In this talk we will survey results on the function $\operatorname{ex}(n,H,F)$ including a number of recent papers. We will also discuss this function’s connection to hypergraph Turán problems.