기초과학연구원 이산수학그룹
Given a $k$-uniform hypergraph $H$, the Ramsey number $R(H;q)$ is the smallest integer $N$ such that any $q$-coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. When $H$ is a complete hypergraph, a classical argument of Erdős, Hajnal, and Rado reduces the general problem to the …
A graph $G$ is (list, DP) $k$-critical if the (list, DP) chromatic number is $k$ but for every proper subgraph $G'$ of $G$, the (list, DP) chromatic number of $G'$ is less than $k$. For $k\geq 4$, we show a bound on the minimum number of edges in a DP $k$-critical graph, and our bound …
Robertson and Seymour introduced branch-width as a connectivity invariant of graphs in their proof of the Wagner conjecture. Decompositions based on this invariant provide a natural framework for implementing dynamic-programming algorithms to solve graph optimization problems. We will discuss the computational issues involved in using branch-width as as a general tool in discrete optimization.
Vertex Cover is perhaps the most-studied problem in parameterized complexity that frequently serves as a testing ground for new concepts and techniques. In this talk, I will focus on a generalization of Vertex Cover called Component Order Connectivity (COC). Given a graph G, an integer k and a positive integer d, the task is to …
We consider a continuous model of graphs, introduced by Dearing and Francis in 1974, where each edge of G to be a unit interval, giving rise to an infinite metric space that contains not only the vertices of G but all points on all edges of G. Several standard graph problems can be defined and …
Given a $k$-uniform hypergraph $F$, its Turán density $\pi(F)$ is the infimum over all $d\in $ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $d\binom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $k\geq 3$, even for small hypergraphs. …
Consider a general Turan-type problem on hypergraphs. Let $\mathcal{F}$ be a family of $k$-subsets of $$ that does not contain sets $F_1, \ldots, F_s$ satisfying some property $P$. We show that if $P$ is low-dimensional in some sense (e.g., is defined by intersections of bounded size) then, under polynomial dependencies between $n, k$ and the …
Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq $ such that the equation \ has no solution with $a_1,a_2,\ldots,a_k\in A$ and integer $x$. Erdős, Sárközy and T. Sós studied $F_{k,2}$, and gave bounds when $k=2,3,4,6$ and also in the general case. We study the problem for $d=3$, and provide bounds for $k=2,3,4,6$ and …
The 5th East Asia Workshop on Extremal and Structural Graph Theory is a workshop to bring active researchers in the field of extremal and structural graph theory, especially in the East Asia such as China, Japan, and Korea. Date November 27, 2025 Thursday (Arrival Day) -- November 30, 2025 Sunday (Departure Day) Venue Fraser Place …
Extremely sparse random binary matrices tend to be singular, due to the likely presence of "local combinatorial dependencies" such as all-zero columns or pairs of identical columns. We discuss this phenomenon, and some results showing that these kinds of combinatorial dependencies are in some sense the "only" causes of singularity. This is joint work with …