Huy Tuan Pham, Random Cayley graphs and Additive combinatorics without groups
2025-01-03 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
A major goal of additive combinatorics is to understand the structures of subsets A of an abelian group G which has a small doubling K = |A+A|/|A|. Freiman’s celebrated theorem first provided a structural characterization of sets with small doubling over the integers, and subsequently Ruzsa in 1999 proved an analog for abelian groups with bounded exponent. Ruzsa further conjectured the correct quantitative dependence on the doubling K in his structural result, which has attracted several interesting developments over the next two decades. I will discuss a complete resolution of (a strengthening of) Ruzsa’s conjecture.
Surprisingly, our approach is crucially motivated by purely graph-theoretic insights, where we find that the group structure is superfluous and can be replaced by much more general combinatorial structures. Using this general approach, we also obtain combinatorial and nonabelian generalizations of classical results in additive combinatorics, and solve longstanding open problems around Cayley graphs and random Cayley graphs motivated by Ramsey theory, information theory and computer science.
Based on joint work with David Conlon, Jacob Fox and Liana Yepremyan.

Tony Huynh, The Peaceable Queens Problem
2025-01-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The peaceable queens problem asks to determine the maximum number $a(n)$ such that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an $n times n$ chessboard so that no queen can capture any queen of the opposite color.
We consider the peaceable queens problem and its variant on the toroidal board. For the regular board, we show that $a(n) leq 0.1716n^2$, for all sufficiently large $n$. This improves on the bound $a(n) leq 0.25n^2$ of van Bommel and MacEachern.
For the toroidal board, we provide new upper and lower bounds. Somewhat surprisingly, our bounds show that there is a sharp contrast in behaviour between the odd torus and the even torus. Our lower bounds are given by explicit constructions. For the upper bounds, we formulate the problem as a non-linear optimization problem with at most 100 variables, regardless of the size of the board. We solve our non-linear program exactly using modern optimization software.
We also provide a local search algorithm and a software implementation which converges very rapidly to solutions which appear optimal. Our algorithm is sufficiently robust that it works on both the regular and toroidal boards. For example, for the regular board, the algorithm quickly finds the so-called Ainley construction.
This is joint work with Katie Clinch, Matthew Drescher, and Abdallah Saffidine.

Laure Morelle, Bounded size modifications in time $2^{{sf poly}(k)}cdot n^2$
2025-01-21 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A replacement action is a function $mathcal L$ that maps each graph to a collection of subgraphs of smaller size.
Given a graph class $mathcal H$, we consider a general family of graph modification problems, called “$mathcal L$-Replacement to $mathcal H$”, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in ${mathcal L}(H_1)$ so that the resulting graph belongs to $mathcal H$.
“$mathcal L$-Replacement to $mathcal H$” can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc.
We prove here that, for any minor-closed graph class $mathcal H$ and for any action $mathcal L$ that is hereditary, there is an algorithm that solves “$mathcal L$-Replacement to $mathcal H$” in time $2^{{sf poly}(k)}cdot |V(G)|^2$, where $sf poly$ is a polynomial whose degree depends on $mathcal H$.

IBS-DIMAG Winter School on Graph Minors, Week 1
2025-02-03 - 2025-02-05 | Room B109, IBS (기초과학연구원)

Jang Soo Kim (김장수), Longest elements in a semigroup of functions and Slater indices
2025-02-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The group ( S_n ) of permutations on ([n]={1,2,dots,n} ) is generated by simple transpositions ( s_i = (i,i+1) ). The length ( ell(pi) ) of a permutation ( pi ) is defined to be the minimum number of generators whose product is ( pi ). It is well-known that the longest element in ( S_n ) has length ( n(n-1)/2 ). Let ( F_n ) be the semigroup of functions ( f:[n]to[n] ), which are generated by the simple transpositions ( s_i ) and the function ( t:[n]to[n] ) given by ( t(1) =t(2) = 1 ) and ( t(i) = i ) for ( ige3 ). The length ( ell(f) ) of a function ( fin F_n ) is defined to be the minimum number of these generators whose product is ( f ). In this talk, we study the length of longest elements in ( F_n ). We also find a connection with the Slater index of a tournament of the
complete graph ( K_n ). This is joint work with Yasuhide Numata.

IBS-DIMAG Winter School on Graph Minors, Week 2
2025-02-10 - 2025-02-12 | Room B109, IBS (기초과학연구원)

Jungho Ahn (안정호), A coarse Erdős-Pósa theorem for constrained cycles
2025-02-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An induced packing of cycles in a graph is a set of vertex-disjoint cycles such that the graph has no edge between distinct cycles of the set. The classic Erdős-Pósa theorem shows that for every positive integer $k$, every graph contains $k$ vertex-disjoint cycles or a set of $O(klog k)$ vertices which intersects every cycle of $G$.
We generalise this classic Erdős-Pósa theorem to induced packings of cycles of length at least $ell$ for any integer $ell$. We show that there exists a function $f(k,ell)=O(ell klog k)$ such that for all positive integers $k$ and $ell$ with $ellgeq3$, every graph $G$ contains an induced packing of $k$ cycles of length at least $ell$ or a set $X$ of at most $f(k,ell)$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$.
Furthermore, we extend the result to long cycles containing prescribed vertices. For a graph $G$ and a set $Ssubseteq V(G)$, an $S$-cycle in $G$ is a cycle containing a vertex in $S$. We show that for all positive integers $k$ and $ell$ with $ellgeq3$, every graph $G$, and every set $Ssubseteq V(G)$, $G$ contains an induced packing of $k$ $S$-cycles of length at least $ell$ or a set $X$ of at most $ell k^{O(1)}$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$.
Our proofs are constructive and yield polynomial-time algorithms, for fixed $ell$, finding either the induced packing of the constrained cycles or the set $X$.
This is based on joint works with Pascal Gollin, Tony Huynh, and O-joung Kwon.

IBS-DIMAG Winter School on Graph Minors, Week 3
2025-02-17 - 2025-02-19 | Room B109, IBS (기초과학연구원)

O-joung Kwon (권오정), Erdős-Pósa property of A-paths in unoriented group-labelled graphs
2025-02-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A family $mathcal{F}$ of graphs is said to satisfy the Erdős-Pósa property if there exists a function $f$ such that for every positive integer $k$, every graph $G$ contains either $k$ (vertex-)disjoint subgraphs in $mathcal{F}$ or a set of at most $f(k)$ vertices intersecting every subgraph of $G$ in $mathcal{F}$. We characterize the obstructions to the Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $Gamma$ and for every subset $Lambda$ of $Gamma$, the family of $Gamma$-labelled $A$-paths whose lengths are in $Lambda$ satisfies the half-integral relaxation of the Erdős-Pósa property. Moreover, we give a characterization of such $Gamma$ and $LambdasubseteqGamma$ for which the same family of $A$-paths satisfies the full Erdős-Pósa property. This is joint work with Youngho Yoo.

Sepehr Hajebi, The pathwidth theorem for induced subgraphs
2025-02-25 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We present a full characterization of the unavoidable induced subgraphs of graphs with large pathwidth. This consists of two results. The first result says that for every forest H, every graph of sufficiently large pathwidth contains either a large complete subgraph, a large complete bipartite induced minor, or an induced minor isomorphic to H. The second result describes the unavoidable induced subgraphs of graphs with a large complete bipartite induced minor.
We will also try to discuss the proof of the first result with as much detail as time permits.
Based on joint work with Maria Chudnovsky and Sophie Spirkl.

Irene Muzi, An elementary bound for Younger's conjecture
2025-03-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 1996, Reed, Robertson, Seymour and Thomas proved Younger’s Conjecture, which states that for all directed graphs D, there exists a function f such that if D does not contain k disjoint cycles, D contains a feedback vertex set, i.e. a subset of vertices whose deletion renders the graph acyclic, of size bounded by f(k). However, the function obtained by Reed, Robertson, Seymour and Thomas is enormous and, in fact, not even elementary. The bound had, so far, not been improved. In this talk, I will present new techniques to improve the bound from non-elementary to elementary. This is joint work with Meike Hatzel, Stephan Kreutzer and Marcelo Milani.

Johannes Carmesin, Open problems in graph theory
2025-03-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Since the proof of the graph minor structure theorem by Robertson and Seymour in 2004, its underlying ideas have found applications in a much broader range of settings than their original context. They have driven profound progress in areas such as vertex minors, pivot minors, matroids, directed graphs, and 2-dimensional simplicial complexes. In this talk, I will present three open problems related to this development, each requiring some background.

Michał Seweryn, Dimension and standard examples in planar posets
2025-03-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The dimension of a poset is the least integer $d$ such that the poset is isomorphic to a subposet of the product of $d$ linear orders. In 1983, Kelly constructed planar posets of arbitrarily large dimension. Crucially, the posets in his construction involve large standard examples, the canonical structure preventing a poset from having small dimension. Kelly’s construction inspired one of the most challenging questions in dimension theory: are large standard examples unavoidable in planar posets of large dimension? We answer the question affirmatively by proving that every $d$-dimensional planar poset contains a standard example of order $Omega(d)$. More generally, we prove that every poset from Kelly’s construction appears in every poset with a planar cover graph of sufficiently large dimension.
joint work with Heather Smith Blake, Jędrzej Hodor, Piotr Micek, and William T. Trotter.

Hyunwoo Lee (이현우), Reconstructing hypergraph matching polynomials
2025-04-01 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
By utilizing the recently developed hypergraph analogue of Godsil’s identity by the second author, we prove that for all $n geq k geq 2$, one can reconstruct the matching polynomial of an $n$-vertex $k$-uniform hypergraph from the multiset of all induced sub-hypergraphs on $lfloor frac{k-1}{k}n rfloor + 1$ vertices. This generalizes the well-known result of Godsil on graphs in 1981 to every uniform hypergraph. As a corollary, we show that for every graph $F$, one can reconstruct the number of $F$-factors in a graph under analogous conditions. We also constructed examples that imply the number $lfloor frac{k-1}{k}n rfloor + 1$ is the best possible for all $ngeq k geq 2$ with $n$ divisible by $k$. This is joint work Donggyu Kim.

Marcelo Garlet Milani, Cycles of Well-Linked Sets and an Elementary Bound for the Directed Grid Theorem
2025-04-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem. The theorem states the existence of a function f such that every digraph of directed tree-width f(k) contains a cylindrical grid of order k as a butterfly minor, but the given function grows non-elementarily with the size of the grid minor.
We present an alternative proof of the directed grid theorem, first proven by Kawarabayashi and Kreutzer in 2015, which is conceptually much simpler, more modular in its composition and also improves the upper bound for the function f to a power tower of height 22.
Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy, who proved a polynomial bound for the excluded grid theorem for undirected graphs. We translate a key concept of their proof to directed graphs by introducing cycles of well-linked sets (CWS), and show that any digraph of high directed tree-width contains a large CWS, which in turn contains a large cylindrical grid.
This is joint work with Meike Hatzel, Stephan Kreutzer and Irene Muzi.

Daniel McGinnis, A necessary and sufficient condition for $k$-transversals
2025-04-14 11:00 - 12:00 | Room B332, IBS (기초과학연구원)
We solve a long-standing open problem posed by Goodman and Pollack in 1988 by establishing a necessary and sufficient condition for a finite family of convex sets in $mathbb{R}^d$ to admit a $k$-transversal (a $k$-dimensional affine subspace that intersects each set) for any $0 le k le d-1$. This result is a common generalization of Helly’s theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$).
This is joint work with Nikola Sadovek.

Nicola Lorenz, A Minor Characterisation of Normally Spanned Sets of Vertices
2025-04-15 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A rooted spanning tree of a graph $G$ is called normal if the endvertices of all edges of $G$ are comparable in the tree order. It is well known that every finite connected graph has a normal spanning tree (also known as depth-first search tree). Also, all countable graphs have normal spanning trees, but uncountable complete graphs for example do not. In 2021, Pitz proved the following characterisation for graphs with normal spanning trees, which had been conjectured by Halin: A connected graph $G$ has a normal spanning tree if and only if every minor of $G$ has countable colouring number, i.e. there is a well-order of the vertices such that every vertex is preceded by only finitely many of its neighbours.
More generally, a not necessarily spanning tree in $G$ is called normal if for every path $P$ in $G$ with both endvertices in $T$ but no inner vertices in $T$, the endvertices of $P$ are comparable in the tree order. We establish a local version of Pitz’s theorem by characterising for which sets $U$ of vertices of $G$ there is a normal tree in $G$ covering $U$. The results are joint work with Max Pitz.

Marcin Briański, Burling Graphs as (almost) universal obstacles to $chi$-boundedness
2025-04-22 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
What causes a graph to have high chromatic number? One obvious reason is containing a large clique (a set of pairwise adjacent vertices). This naturally leads to investigation of (chi)-bounded classes of graphs — classes where a large clique is essentially the only reason for large chromatic number.
Unfortunately, many interesting graph classes are not (chi)-bounded. An eerily common obstruction to being (chi)-bounded are the Burling graphs — a family of triangle-free graphs with unbounded chromatic number. These graphs have served as counterexamples in many settings: demonstrating that graphs excluding an induced subdivision of (K_{5}) are not (chi)-bounded, that string graphs are not (chi)-bounded, that intersection graphs of boxes in ({mathbb{R}}^{3}) are not (chi)-bounded, and many others.
In many of these cases, this sequence is the only known obstruction to (chi)-boundedness. This led Chudnovsky, Scott, and Seymour to conjecture that any graph of sufficiently high chromatic number must either contain a large clique, an induced proper subdivision of a clique, or a large Burling graph as an induced subgraph.
The prevailing belief was that this conjecture should be false. Somewhat surprisingly, we did manage to prove it under an extra assumption on the “locality” of the chromatic number — that the input graph belongs to a (2)-controlled family of graphs, where a high chromatic number is always certified by a ball of radius (2) with large chromatic number.
In this talk, I will present this result and discuss its implications in structural graph theory, and algorithmic implications to colouring problems in specific graph families.
This talk is based on joint work with Tara Abrishami, James Davies, Xiying Du, Jana Masaříková, Paweł Rzążewski, and Bartosz Walczak conducted during the STWOR2 workshop in Chęciny Poland.

Pascal Schweitzer, Recent insights surrounding combinatorial approaches to isomorphism and symmetry problems
2025-04-28 11:00 - 12:00 | Room B332, IBS (기초과학연구원)
Modern practical software libraries that are designed for isomorphism tests and symmetry computation rely on combinatorial techniques combined with techniques from algorithmic group theory. The Weisfeiler-Leman algorithm is such a combinatorial technique. When taking a certain view from descriptive complexity theory, the algorithm is universal. After an introduction to problems arising in symmetry computation and this particular combinatorial technique, I will give an overview of results from recent years. The results give insights into worst-case behavior and computational complexity.

Eunjin Oh (오은진), Approximation Algorithms for the Geometric Multimatching Problem
2025-04-29 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let S and T be two sets of points in a metric space with a total of n points. Each point in S and T has an associated value that specifies an upper limit on how many points it can be matched with from the other set. A multimatching between S and T is a way of pairing points such that each point in S is matched with at least as many points in T as its assigned value, and vice versa for each point in T. The cost of a multimatching is defined as the sum of the distances between all matched pairs of points. The geometric multimatching problem seeks to find a multimatching that minimizes this cost. A special case where each point is matched to at most one other point is known as the geometric many-to-many matching problem.
We present two results for these problems when the underlying metric space has a bounded doubling dimension. Specifically, we provide the first near-linear-time approximation scheme for the geometric multimatching problem in terms of the output size. Additionally, we improve upon the best-known approximation algorithm for the geometric many-to-many matching problem, previously introduced by Bandyapadhyay and Xue (SoCG 2024), which won the best paper award at SoCG 2024.
This is joint work with Shinwoo An and Jie Xue.

Seokbeom Kim (김석범), The structure of △(1, 2, 2)-free tournaments
2025-05-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known.
Let $triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $triangle(1, 2, 2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments.
This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.

Meike Hatzel, Counterexample to Babai's lonely colour conjecture
2025-05-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Motivated by colouring minimal Cayley graphs, in 1978 Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge colouring in which each cycle contains no colour exactly once.
The result presented is the joint work with James Davies and Liana Yepremyan.

Denys Bulavka, Strict Erdős-Ko-Rado Theorems for Simplicial Complexes
2025-06-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The now classical theorem of Erdős, Ko and Rado establishes
the size of a maximal uniform family of pairwise-intersecting sets as well as a characterization of the families attaining such upper bound. One natural extension of this theorem is that of restricting the possiblechoices for the sets. That is, given a simplicial complex, what is the size of a maximal uniform family of pairwise-intersecting faces. Holroyd and Talbot, and Borg conjectured that the same phenomena as in the classical case (i.e., the simplex) occurs: there is a maximal size pairwise-intersecting family with all its faces having some common vertex. Under stronger hypothesis, they also conjectured that if a family attains such bound then its members must have a common vertex. In this talk I will present some progress towards the characterization of the maximal families. Concretely I will show that the conjecture is true for near-cones of sufficiently high depth. In particular, this implies that the characterization of maximal families holds for, for example, the independence complex of a chordal graph with an isolated vertex as well as the independence complex of a (large enough) disjoint union of graphs with at least one isolated vertex. Under stronger hypothesis, i.e., more isolated vertices, we also recover a stability theorem.
This talk is based on a joint work with Russ Woodroofe.

On-Hei Solomon Lo, Minors of non-hamiltonian graphs
2025-06-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A seminal result of Tutte asserts that every 4-connected planar graph is hamiltonian. By Wagner’s theorem, Tutte’s result can be restated as: every 4-connected graph with no $K_{3,3}$ minor is hamiltonian. In 2018, Ding and Marshall posed the problem of characterizing the minor-minimal 3-connected non-hamiltonian graphs. They conjectured that every 3-connected non-hamiltonian graph contains a minor of $K_{3,4}$, $mathfrak{Q}^+$, or the Herschel graph, where $mathfrak{Q}^+$ is obtained from the cube by adding a new vertex and connecting it to three vertices that share a common neighbor in the cube. We recently resolved this conjecture along with some related problems. In this talk, we review the background and discuss the proof.

Attila Jung, The Quantitative Fractional Helly Theorem
2025-06-17 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Two celebrated extensions of Helly’s theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Barany, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $F$ of $n$ convex sets in $mathbb{R}^d$ such that at least $alpha binom{n}{d+1}$ of the $(d+1)$-tuples of $F$ have an intersection of volume at least 1, then one can select $Omega_{d,alpha}(n)$ members of $F$ whose intersection has volume at least $Omega_d(1)$. Joint work with Nora Frankl and Istvan Tomon.

Roohani Sharma, Uniform and Constructive Polynomial Kernel for Deletion to $K_{2,p}$ Minor-Free Graphs
2025-06-25 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let $mathcal F$ be a fixed, finite family of graphs. In the $mathcal F$-Deletion problem, the input is a graph G and a positive integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of $mathcal F$ as a minor. The $mathcal F$-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-$eta$ Deletion, Treedepth-$eta$ Deletion, Pathwidth-$eta$ Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the $mathcal F$-Deletion problem from the kernelization perspective.
In a seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family $mathcal F$: that is, the size of the kernel is $k^{f(mathcal F)}$, for some function f that depends only on $mathcal F$. Also the size of the kernel depends on non-constructive constants.
Later Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] showed that the non-uniformity in the kernel size bound of Fomin et al. is unavoidable as Treewidth-$eta$ Deletion, cannot admit a kernel of size $O(k^{(eta +1)/2 -epsilon)}$, for any $epsilon >0$, unless NP $subseteq$ coNP/poly. On the other hand it was also shown that Treedepth-$eta$ Deletion, admits a uniform kernel of size $f(mathcal F) k^6$, showcasing that there are subclasses of $mathcal F$ where the asymptotic kernel sizes do not grow as a function of the family $mathcal F$. This work leads to the natural question of determining classes of $mathcal F$ where the problem admits uniform polynomial kernels.
In this work, we show that if all the graphs in $mathcal F$ are connected and $mathcal F$ contains $K_{2,p}$ (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size $f(mathcal F) k^{10}$ where the constants in the size bound are also constructive. The graph $K_{2,p}$ is one natural extension of the graph $theta_p$, where $theta_p$ is a graph on two vertices and p parallel edges. The case when $mathcal F$ contains $theta_p$ has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.
This is joint work with William Lochet.

Sergey Norin, Asymptotic dimension of intersection graphs
2025-07-01 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The notion of asymptotic dimension of metric spaces, introduced by Gromov, describes their large-scale behaviour. Asymptotic dimension of graph families has been recently studied, in particular, by Bonamy et al. who proved that the asymptotic dimension of proper minor-closed graph families is at most two.
We will discuss nerve-type theorems for asymptotic dimension. In particular, we show that the asymptotic dimension of intersection graphs of balls and spheres in $mathbb{R}^d$ is at most $d+1$.
Based on joint work with Zdeněk Dvořák and with Chun-Hung Liu.

Mihyun Kang (강미현), Phase transitions in a random subgraph of the hypercube
2025-07-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will discuss classical and recent results about phase transitions in random subgraphs of the hypercube and beyond. The focus will be on the giant component, long cycles, large matchings, and isoperimetric properties.

2025 Summer School on Combinatorics and Algorithms (2025 조합론 및 알고리즘 여름학교)
2025-07-14 14:00 - 2025-07-18 17:00 | POSTECH, Pohang, Korea
The 2025 Summer School on Combinatorics and Algorithms is a venue for students and early-career researchers to learn selected topics in theoretical computer science and discrete mathematics. It will be a great opportunity for young and aspiring researchers to study topics which are important but not covered during the lectures in the university classes.
Website: https://combialgo.dimag.kr/
Lecturers and Topics

Édouard Bonnet, LIP, CNRS.

Induced Subgraphs and Induced Minors of Graphs

Michał Pilipczuk, University of Warsaw.

Structural Theory of Sparse Graphs

Schedule

Start on 14 July 2025 Monday, 2 PM
End on 18 July 2025 Friday, 5 PM

Linda Cook, A tight algorithmic meta-theorem for distributed certification within bounded treewidth graphs
2025-07-22 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A local certification of a graph property is a protocol in which nodes are given “certificates of a graph property” that allow the nodes to check whether their network has this property while only communicating with their local network. The key property of a local certification is that if certificates are corrupted, some node in the network will be able to recognize this. Inspired by practical concerns, the aim in LOCAL certification is to minimize the maximum size of a certificate.
In this talk we introduce local certification and open problems in the area and present some recent joint work with Eunjung Kim and Tomáš Masařík, A Tight Meta-theorem for LOCAL Certification of MSO2 Properties within Bounded Treewidth Graphs.
In this work, instead of considering a specific graph property and developing a local certification protocol tailor-made for this property, we aim for generic protocols that can certify any property expressible in a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Omega(n^2/log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. Inspired by their importance in centralized computing and Robertson-Seymour Graph Minor theory, we consider graphs of bounded treewidth. We provide a local certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a local certification protocol for certifying bounded treewidth. That is, for each integer $k$ and each MSO$_2$-expressible property $Pi$, we give a local certification protocol to certify that a graph satisfies $Pi$ and has treewidth at most $k$ using certificates of size $mathcal{O}(log n)$ (which is asymptotically optimal). Our result improves upon the works of Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang (PODC 2025).
