기초과학연구원 이산수학그룹
The strong product $G\boxtimes H$ of graphs $G$ and $H$ is the graph on the cartesian product $V(G)\times V(H)$ such that vertices $(v,w)$ and $(x,y)$ are adjacent if and only if $\max\{d_G(v,x),d_H(w,y)\}=1$. Graph product structure theory aims to describe complicated graphs in terms of subgraphs of strong products of simpler graphs. This area of research was initiated …
Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. …
Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. …
A subset of a group is said to be product free if it does not contain the product of two elements in it. We consider how large can a product free subset of $A_n$ be? In the talk we will completely solve the problem by determining the largest product free subset of $A_n$. Our proof …
We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this talk, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order …
We confirm a conjecture of Gartland and Lokshtanov : if for a hereditary graph class $\mathcal{G}$ there exists a constant $k$ such that no member of $\mathcal{G}$ contains a $k$-creature as an induced subgraph or a $k$-skinny-ladder as an induced minor, then there exists a polynomial $p$ such that every $G \in \mathcal{G}$ contains at …
We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that, given a directed graph G, two integers $s,t\in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$, with probability $2^{−\operatorname{poly}(k)}$ the set $Z$ …
Random walks and spectral methods have had a strong influence on modern graph algorithms as evidenced by the extensive literature on the subject. In this talk, I will present how random walks helped make progress on algorithmic problems on planar graphs. In particular, I show how random walk based (i.e., spectral) approaches led to progress …
Property testers are probabilistic algorithms aiming to solve a decision problem efficiently in the context of big-data. A property tester for a property P has to decide (with high probability correctly) whether a given input graph has property P or is far from having property P while having local access to the graph. We study …
A temporal graph is a graph whose edges are available only at specific times. In this scenario, the only valid walks are the ones traversing adjacent edges respecting their availability, i.e. sequence of adjacent edges whose appearing times are non-decreasing. Given a graph G and vertices s and t of G, Menger’s Theorem states that …