기초과학연구원 이산수학그룹
Let $\mathcal{F}$ be a family of graphs, and let $p$ and $r$ be nonnegative integers. The $(p,r,\mathcal{F})$-Covering problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where …
By a seminal result of Valiant, computing the permanent of (0, 1)-matrices is, in general, #P-hard. In 1913 Pólya asked for which (0, 1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the …
A linear $3$-graph is called a (3-)hypertree if there exists exactly one path between each pair of two distinct vertices. A linear $3$-graph is called a Steiner triple system if each pair of two distinct vertices belong to a unique edge. A simple greedy algorithm shows that every $n$-vertex Steiner triple system $G$ contains all …
The disjoint paths logic, FOL+DP, is an extension of First Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in \{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every …
Various types of independent sets have been studied for decades. As an example, the minimum number of maximal independent sets in a connected graph of given order is easy to determine (hint; the answer is written in the stars). When considering this question for twin-free graphs, it becomes less trivial and one discovers some surprising …
The Graphic Travelling Salesman Problem is the problem of finding a spanning closed walk (a TSP walk) of minimum length in a given connected graph. The special case of the Graphic TSP on subcubic graphs has been studied extensively due to their worst-case behaviour in the famous $\frac{4}{3}$-integrality-gap conjecture on the "subtour elimination" linear programming …
I will first introduce the notion of recognisability of languages of terms and then its extensions to sets of relational structures. In a second step, I will discuss relations with decompositions of graphs/matroids and why their MSOL-definability is related to understanding recognisable sets. I will finally explain how to define in MSOL branch-decompositions for finitely …
We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated by twin-width (or simply, delineated) if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for …
The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will sketch a proof of this conjecture for every large n. Joint work with D. Kang, T. Kelly, D. Kühn and D. Osthus.
Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S. We prove this conjecture in the first open case of t=4. …