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.