Discrete Mathematics Group

Amadeus Reinald, Oriented trees in $O(k sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture
2024-09-03 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $frac{k^2}{2} – frac{k}{2} + 1$ by Addario-Berry et al.
In this talk, we give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8sqrt{frac{2}{15}} k sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $sqrt{frac{4}{3}} k sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $bge 2$.

Neal Bushaw, Edge-colored Extremal Problems
2024-09-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An edge-colored graph $H$ is called rainbow if all of its edges are given distinct colors. An edge-colored graph $G$ is then called rainbow $H$-free when no copy of $H$ in $G$ is rainbow. With this, we define a graph $G$ to be rainbow $H$-saturated when there is some proper edge-coloring of $G$ which is rainbow $H$-free, but for every pair of non-adjacent vertices $x,yin V(G)$, the graph $G+xy$ formed by adding the edge $xy$ to $G$ cannot be given a rainbow $H$-free coloring. We think of these graphs as edge-maximal rainbow $H$-free graphs. (We note that here we make no restrictions on the colorings of $G+xy$ whatsoever, except that they are proper colorings. They may use any number of colors, and need not be extensions of the original rainbow $H$-free coloring of $G$.)

With this framework in place, we define the rainbow saturation number and rainbow extremal number to be the largest and smallest number of edges, respectively, among all $n$ vertex rainbow $H$-saturated graphs. The latter of these was defined by Keevash, Mubayi, Sudakov, and Verstraëte in 2007; the former was introduced by B., Johnston, and Rombach in 2019.

In this talk, we discuss recent progress on both the rainbow saturation numbers and rainbow extremal numbers. We also give several broad generalizations of these concepts and discuss many open problems. This talk contains joint work with Vic Bednar (Furman), Dan Johnston (Trinity College, CT), and Puck Rombach (Vermont).

Zhihan Jin (金之涵), The Helly number of Hamming balls and related problems
2024-09-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove the following variant of Helly’s classical theorem for Hamming balls with a bounded radius. For $n > t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X| > 1$ and all $n > t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory. This is joint work with N. Alon and B. Sudakov.

Gábor Tardos, Extremal theory of 0-1 matrices
2024-09-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We say that a 0-1 matrix A contains another such matrix (pattern) P if P can be obtained from a submatrix of A by possibly changing a few 1 entries to 0. The main question of this theory is to estimate the maximal number of 1 entries in an n by n 0-1 matrix NOT containing a given pattern P. This question has very close connections to Turan type extremal graph theory and also to the Devenport-Schinzel theory of sequences. Results in the extremal theory of 0-1 matrices proved useful in attacking problems in far away fields as combinatorial geometry and the analysis of algorithms.
This talk will concentrate on acyclic patterns and survey some old and recent results in the area and will also contain several open problems.

Mathias Schacht, Canonical colourings in random graphs
2024-10-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $ellgeq 2$ they showed that $n^{-frac{2}{ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_ell$.
We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_ell$ can be ensured in the deterministic setting.
We transfer the Erdős-Rado theorem to the random environment and show that for $ellgeq 4$ both thresholds coincide. As a consequence the proof yields $K_{ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_ell$.
This is joint work with Nina Kamčev.

Kyeongsik Nam (남경식), Random walks on percolation
2024-10-15 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

Colin Geniet, Permutations, patterns, and twin-width
2024-10-22 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
This talk will first introduce combinatorics on permutations and patterns, presenting the basic notions and some fundamental results: the Marcus-Tardos theorem which bounds the density of matrices avoiding a given pattern, and the Guillemot-Marx algorithm for pattern detection using the notion now known as twin-width.
I will then present a decomposition result: permutations avoiding a pattern factor into bounded products of separable permutations. This can be rephrased in terms of twin-width: permutation with bounded twin-width are build from a bounded product of permutations of twin-width 0. Comparable results on graph encodings follow from this factorisation.
This is joint work with Édouard Bonnet, Romain Bourneuf, and Stéphan Thomassé.

Felix Christian Clemen, Triangles in the Plane
2024-10-29 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of $n$ points in the plane. Since then a great variety of extremal problems in finite planar point sets have been studied. Here, we look at such questions concerning triangles. Among others we answer the following question asked by Erdős and Purdy almost 50 years ago: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles?
For our proofs we use hypergraph Turán theory. This is joint work with Balogh and Dumitrescu.

Michał Pilipczuk, Monadic stability and monadic dependence
2024-11-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will give an overview of the recent attempts of building a structure theory for graphs centered around First-Order transductions: a notion of containment inspired by finite model theory. Particularly, we will speak about the notions of monadic dependence and monadic stability, their combinatorial characterizations, and the developments on the algorithmic front.

IBS-DIMAG Workshop on Topology and Combinatorics
2024-11-11 11:00 - 17:30 | Room B332, IBS (기초과학연구원)
The IBS-DIMAG Workshop on Topology and Combinatorics will be held on November 11, 2024 at Room B332, Institute for Basic Science (IBS), Daejeon, South Korea.
Invited Speakers (tentative)

Karim Adiprasito (Jussieu Institute of Mathematics)
Minho Cho조민호 (IBS Extremal Combinatorics and Probability Group)
Niloufar Fuladi (INRIA Center of Université de Lorraine)
Minki Kim김민기 (GIST)
Dohyeon Lee이도현 (KAIST & IBS Discrete Mathematics Group)
Geunho Lim임근호 (Jussieu Institute of Mathematics)
Semin Yoo유세민 (IBS Discrete Mathematics Group)

Tentative Schedule

11:00-11:30 Karim Adiprasito, The Charney Davis conjecture and Boolean decompositions
12:00-14:00 Lunch Break
14:00-14:30 Niloufar Fuladi, Universal families of arcs and curves on surfaces
14:30-15:00 Semin Yoo유세민, The Charney-Davis conjecture and its related complexes (survey)
15:00-15:30 Geunho Lim임근호, Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
15:30-16:00 Break
16:00-16:30 Minki Kim김민기, Some extensions of the colorful Helly theorem
16:30-17:00 Minho Cho조민호, Fractional Helly theorems via weak saturation
17:00-17:30 Dohyeon Lee이도현, Betti number of clique complex of H-free graphs (survey)
18:00-20:00 Banquet

Organizer

Semin Yoo유세민, IBS Discrete Mathematics Group

Abstracts
Karim Adiprasito, The Charney Davis conjecture and Boolean decompositions
I will present a novel approach to the Charney Davis conjecture by introducing anticommutative Lefschetz properties, and studying them on the Danzer complex of flag spheres.
Semin Yoo, The Charney-Davis conjecture and its related complexes (survey)
The Charney-Davis conjecture lies in the intersection between topology and combinatorics. The conjecture is a special case of the Hopf-Chern-Thurston conjecture for all nonpositively curved, piecewise Euclidean manifolds which are cellulated by regular Euclidean cubes. In 2001, Davis and Okun solved the conjecture for a 4-dimensional case, and some other partial results are also known. However, the conjecture is still open so far.
In this talk, I will describe five statements that solve the Charney-Davis conjecture and explain some complexes to achieve this goal.
Niloufar Fuladi, Universal families of arcs and curves on surfaces
In this talk I introduce a family of curves that realize all pants decomposition of a surface: a family of simple closed curves Γ on a surface realizes all types of pants decompositions if for any pants decomposition of the surface, there exists a homeomorphism sending it to a subset of the curves in Γ. The study of such universal families of curves is motivated by questions on graph embeddings, joint crossing numbers and finding an elusive center of moduli space. I will discuss the case of surfaces without punctures, where we establish an exponential upper bound and a superlinear lower bound on the minimum size of the family of curves with this universal property. I also talk about a similar concept of universality for triangulations of polygons, where we provide bounds which are tight up to logarithmic factors. In this talk, I will explain the background and context and no prior knowledge in topology or surfaces is needed.
This is joint work with Arnaud de Mesmay and Hugo Parlier.
Geunho Lim, Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on manifolds with arbitrary fundamental groups, as a generalization of the Atiyah-Singer rho-invariant. There are many interesting applications in geometry, topology, and combinatorics. In this talk, we show linear bounds on the rho-invariants in terms of simplicial complexity of manifolds by using hyperbolization methods. As applications, we give new concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces. This is a joint work with Shmuel Weinberger.
Minki Kim, Some extensions of the colorful Helly theorem
Given $k > d$ finite families of convex sets in $mathbb{R}^d$, if every colorful $(d+1)$-tuple is intersecting, then the union of $k-d$ color classes is intersecting. This is known as the colorful Helly theorem. Based on recent joint work with Alan Lew, I will present some extensions of the colorful Helly theorem and their applications.
Minho Cho, Fractional Helly theorems via weak saturation
Two celebrated extensions of the classical Helly’s theorem are the fractional Helly theorem and the colorful Helly theorem. Bulavka, Goodarzi, and Tancer recently established the optimal bound for the unified generalization of the fractional and the colorful Helly theorems using a colored extension of the exterior algebra. In this talk, we introduce a combinatorial reduction of both the fractional Helly theorem and its colorful version to a classical problem in extremal combinatorics known as weak saturation. No such results connecting the fractional Helly theorem and weak saturation are known in the long history of literature. These reductions, along with basic linear algebraic arguments for the reduced weak saturation problems, let us give new short proofs of the optimal bounds for both the fractional Helly theorem and its colorful version without using exterior algebra.
This is joint work with Debsoumya Chakraborti, Jinha Kim, and Minki Kim.
Dohyeon Lee, Betti number of clique complex of H-free graphs (survey)
The clique complex of a graph is a simplicial complex whose faces correspond to cliques of the given graph. In this talk, we investigate extremal properties of the Betti numbers of clique complexes. While general simplicial complexes with n vertices can have a total Betti number approximately $2^n$, Adamaszek showed that Betti number of clique complex is bounded by around $1.32^n$. Furthermore, if the underlying graph has an independence number at most two, the bound tightens to $1.25^n$. Adiprasito, Nevo, and Tancer established additional bounds on the total Betti number for clique complexes of (induced) H-free graphs for various graph H, along with constructions of graphs with large total Betti number. In another direction, Zhang and Wu demonstrated that for ternary graphs (whose with no induced cycles of length 3n), the Betti number of the independence complex (the clique complex of the complement of a graph) is at most one. Indeed, Jinha Kim showed that these independence complexes are either contractible or homotopy spheres.

Karim Adiprasito, Ehrhart theory revisited: Algebraic aspects, unimodality and more
2024-11-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Ehrhart theory is the study of lattice polytopes, specifically aimed at understanding how many lattice points are inside dilates of a given lattice polytope, and the study has a wide range of connections ranging from coloring graphs to mirror symmetry and representation theory. Recently, we introduced new algebraic tools to understand this theory, and resolve some classical conjectures. I will explain the combinatorial underpinnings behind two of the key techniques: Parseval identities for semigroup algebras, and the character algebra of a semigroup.

Eng Keat Hng, Graphon branching processes and fractional isomorphism
2024-11-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 2005, Bollobás, Janson and Riordan introduced and extensively studied a general model of inhomogeneous random graphs parametrised by graphons. In particular, they studied the emergence of a giant component in these inhomogeneous random graphs by relating them to a broad collection of inhomogeneous Galton-Watson branching processes.
Fractional isomorphism of finite graphs is an important and well-studied concept at the intersection of graph theory and combinatorial optimisation. It has many different characterizations that involve a range of very different and seemingly unrelated properties of graphs. Recently, Grebík and Rocha developed a theory of fractional isomorphism for graphons.
In our work, we characterise inhomogeneous random graphs that yield the same inhomogeneous Galton-Watson branching process (and hence have a similar component structure).
This is joint work with Jan Hladký and Anna Margarethe Limbach.

Yulai Ma, Pairwise disjoint perfect matchings in regular graphs
2024-12-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter.
For ${0 leq lambda leq r}$, let $m(lambda,r)$ denote the maximum number $s$ such that every $lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(lambda,r)$ are known only in limited cases; for example, $m(3,3)=m(4,r)=1$, and $m(r,r) leq r-2$ for all $r
ot = 5$, with $m(r,r) leq r-3$ when $r$ is a multiple of $4$. In this talk, we present new upper bounds for $m(lambda,r)$ and examine connections between $m(5,5)$ and several well-known conjectures for cubic graphs.
This is joint work with Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf.

Jun Gao (高峻), Phase transition of degenerate Turán problems in p-norms
2024-12-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a positive real number $p$, the $p$-norm $|G|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We determine the exact value of $p_{F}$, under a mild assumption on the growth rate of $mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well.
We also prove a general upper bound that is tight up to a $log n$ factor for $mathrm{ex}_{p}(n,F)$ when $p = p_{F}$.
We conjecture that this $log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.
This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

Joonkyung Lee (이준경), Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
2024-12-17 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex colourings.
We obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular, we prove that, for any antiferromagnetic $G$,
$|mathrm{Hom}(K_d, G)|^{1/d} ≤ |mathrm{Hom}(K_{d,d} setminus M, G)|^{1/(2d)}$
holds, where $K_{d,d} setminus M$ denotes the complete bipartite graph $K_{d,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah, Sawhney, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest.
Joint work with Jaeseong Oh and Jaehyeon Seo.

Zixiang Xu (徐子翔), Multilinear polynomial methods and stability results on set systems
2024-12-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 1966, Kleitman established that if ( |A triangle B| leq d ) for any ( A, B in mathcal{F} ), then ( |mathcal{F}| leq sum_{i=0}^{k} binom{n}{i} ) for ( d = 2k ), and ( |mathcal{F}| leq 2 sum_{i=0}^{k} binom{n-1}{i} ) for ( d = 2k+1 ). These upper bounds are attained by the radius-(k) Hamming ball ( mathcal{K}(n, k) := { F : F subseteq [n], |F| leq k } ) in the even case, and by the family ( mathcal{K}_y(n, k) := { F : F subseteq [n], |F setminus {y}| leq k } ) in the odd case. In 2017, Frankl provided a combinatorial proof of a stability result for Kleitman’s theorem, offering improved upper bounds for ( |mathcal{F}| ) when ( mathcal{F} ) is not the extremal structure.
In this talk, I will begin by demonstrating the application of multilinear polynomial methods in extremal set theory, highlighting some interesting techniques. I will then present an algebraic proof of the stability result for Kleitman’s theorem. Finally, I will discuss further applications and explore how to employ linear algebra methods more effectively and flexibly.
This talk is based on joint work with Jun Gao and Hong Liu.

Huy Tuan Pham, Random Cayley graphs and Additive combinatorics without groups
2025-01-03 16:30 - 17:30 | 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.
