기초과학연구원 이산수학그룹
We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle and it answers the open question of Bergougnoux and Kanté . We also show …
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. …
A circle graph is an intersection graph of a set of chords of a circle. In this talk, I will describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply that treewidth and Hadwiger number are linearly tied on the class …
We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all …
Let $G=\mathbb{F}_p^n$. Which systems of linear equations $\Psi$ have the property that amongst all subsets of $G$ of fixed density, random subsets minimise the number of solutions to $\Psi$? This is an arithmetic analogue of a well-known conjecture of Sidorenko in graph theory, which has remained open and of great interest since the 1980s. We …
Let $X$ be a 2-dimensional simplicial complex. Denote by $\text{ex}_{\hom}(n,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of $\text{ex}_{\hom}(n,X)$ have recently been determined for all surfaces $X$. I will discuss these results, as well as ongoing work bounding $\text{ex}_{\hom}(n,X)$ for arbitrary 2-dimensional …
Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal of the planar disjoint paths problem is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. This problem has been studied extensively due to …
A subset $A\subseteq \mathbb Z$ of integers is free if for every two distinct subsets $B, B'\subseteq A$ we have \Pisier asked if for every subset $A\subseteq \mathbb Z$ of integers the following two statement are equivalent: (i) $A$ is a union of finitely many free sets. (ii) There exists $\epsilon>0$ such that every finite …
We give a summary on the work of the last months related to Frankl's Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $\mathcal F$), then $\mathcal F$ contains an …
In 1977, Erdős and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $|G|^c$ replaced by $2^{c\sqrt{\log |G|}}$. There has no improvement on this result (for general …