Robert Hickingbotham, Powers of planar graphs, product structure, and blocking partitions
2023-10-24 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Graph product structure theory describes complex graphs in terms of products of simpler graphs. In this talk, I will introduce this subject and talk about some of my recent results in this area. The focus of my talk will be on a new tool in graph product structure theory called `blocking partitions.’ I’ll show how this tool can be used to prove stronger product structure theorems for powers of planar graphs as well as $k$-planar graphs, resolving open problems of Dujmović, Morin and Wood, and Ossona de Mendez.

James Davies, Odd distances in colourings of the plane
2023-10-31 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.

The 3rd East Asia Workshop on Extremal and Structural Graph Theory
2023-11-01 - 2023-11-05 | The Southern Beach Hotel & Resort Okinawa
The 3rd East Asia Workshop on Extremal and Structural Graph Theory is a workshop to bring active researchers in the field of extremal and structural graph theory, especially in the East Asia such as China, Japan, and Korea.
Website: http://tgt.ynu.ac.jp/2023EastAsia.html

Bruce A. Reed, Some Variants of the Erdős-Sós Conjecture
2023-11-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph.
We present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.

Seunghun Lee (이승훈), On colorings of hypergraphs embeddable in $mathbb{R}^d$
2023-11-20 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:Vto [m]$ such that every hyperedge of $H$ is not monochromatic. The (weak) chromatic number of $H$, denoted by $chi(H)$, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T subseteq V$ is called a transversal of $H$ if for every hyperedge $e$ of $H$ we have $Tcap e
e emptyset$. The transversal number of $H$, denoted by $tau(H)$, is the smallest size of a transversal in $H$. The transversal ratio of $H$ is the quantity $tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $chi(H)$, these two quantities are closely related to each other.
Upon my previous presentation, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ), we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $mathbb{R}^d$ by varying $k$, $d$, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise, Panagiotou, Pikhurko, and Taraz, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.

Hyunwoo Lee (이현우), Towards a high-dimensional Dirac's theorem
2023-11-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Dirac’s theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles.
We consider another natural generalization of the perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs.
We prove that for sufficiently large integer $n$ with $n equiv 1 text{ or } 3 pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $left(frac{3 + sqrt{57}}{12} + o(1) right)n = (0.879… + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs.
We conjecture that the number $frac{3 + sqrt{57}}{12}$ can be replaced with $frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac’s theorem.

Ben Lund, Almost spanning distance trees in subsets of finite vector spaces
2023-12-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For $dge 2$ and an odd prime power $q$, let $mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field $mathbb{F}_q$. The distance between two points $(x_1,ldots,x_d)$ and $(y_1,ldots,y_d)$ is defined to be $sum_{i=1}^d (x_i-y_i)^2$. An influential result of Iosevich and Rudnev is: if $E subset mathbb{F}_q^d$ is sufficiently large and $t in mathbb{F}_q$, then there are a pair of points $x,y in E$ such that the distance between $x$ and $y$ is $t$. Subsequent works considered embedding graphs of distances, rather than a single distance. I will discuss joint work with Debsoumya Chakraborti, in which we show that every sufficiently large subset of $mathbb{F}_q^d$ contains every nearly spanning tree of distances with bounded degree in each distance. The main novelty in this result is that the distance graphs we find are nearly as large as the set $S$ itself, but even for smaller distance trees our work leads to quantitative improvements to previously known bounds. A key ingredient in our proof is a new colorful generalization of a classical result of Haxell on finding nearly spanning bounded-degree trees in expander graphs. This is joint work with Debsoumya Chakraborti.

Ting-Wei Chao (趙庭偉), Tight Bound on Joints Problem and Partial Shadow Problem
2023-12-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given a set of lines in $mathbb R^d$, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in $mathbb R^3$ via the polynomial method.
Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.

Shengtong Zhang (张盛桐), Triangle Ramsey numbers of complete graphs
2023-12-19 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that [r_{K_3}(K_t)=binom{r(K_t)}3] for all sufficiently large $t$. Our proof involves a combination of results on the chromatic number of triangle-sparse graphs.
Joint work with Jacob Fox and Jonathan Tidor.

Daniel McGinnis, Applications of the KKM theorem to problems in discrete geometry
2024-01-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We present the KKM theorem and a recent proof method utilizing it that has proven to be very useful for problems in discrete geometry. For example, the method was used to show that for a planar family of convex sets with the property that every three sets are pierced by a line, there are three lines whose union intersects each set in the family. This was previously a long-unsolved problem posed by Eckhoff. We go over a couple of examples demonstrating the method and propose a potential future research direction to push the method even further.

Jinyoung Park (박진영), Dedekind's Problem and beyond
2024-01-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Dedekind’s Problem asks the number of monotone Boolean functions, a(n), on n variables. Equivalently, a(n) is the number of antichains in the n-dimensional Boolean lattice $[2]^n$. While the exact formula for the Dedekind number a(n) is still unknown, its asymptotic formula has been well-studied. Since any subsets of a middle layer of the Boolean lattice is an antichain, the logarithm of a(n) is trivially bounded below by the size of the middle layer. In the 1960’s, Kleitman proved that this trivial lower bound is optimal in the logarithmic scale, and the actual asymptotics was also proved by Korshunov in 1980’s. In this talk, we will discuss recent developments on some variants of Dedekind’s Problem. Based on joint works with Matthew Jenssen, Alex Malekshahian, Michail Sarantis, and Prasad Tetali.

Matthew Kroeker, Average flat-size in complex-representable matroids
2024-01-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Melchior’s Inequality (1941) implies that, in a rank-3 real-representable matroid, the average number of points in a line is less than three. This was extended to the complex-representable matroids by Hirzebruch in 1983 with the slightly weaker bound of four. In this talk, we discuss and sketch the proof of the recent result that, in a rank-4 complex-representable matroid which is not the direct-sum of two lines, the average number of points in a plane is bounded above by an absolute constant. Consequently, the average number of points in a flat in a rank-4 complex-representable matroid is bounded above by an absolute constant. Extensions of these results to higher dimensions will also be discussed. In particular, the following quantities are bounded in terms of k and r respectively: the average number of points in a rank-k flat in a complex-representable matroid of rank at least 2k-1, and the average number of points in a flat in a rank-r complex-representable matroid. Our techniques rely on a theorem of Ben Lund which approximates the number of flats of a given rank.
This talk is based on joint work with Rutger Campbell and Jim Geelen.

Zichao Dong, Convex polytopes in non-elongated point sets in $mathbb{R}^d$
2024-01-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For any finite point set $P subset mathbb{R}^d$, we denote by $text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P subset mathbb{R}^d$ in general position, satisfying $text{diam}(P) < alphasqrt[d]{n}$ (informally speaking, `non-elongated’), contains a convex $c$-polytope. Valtr proved that $c_{2, alpha}(n) approx sqrt[3]{n}$, which is asymptotically tight in the plane. We generalize the results by establishing $c_{d, alpha}(n) approx n^{frac{d-1}{d+1}}$. Along the way we generalize the definitions and analysis of convex cups and caps to higher dimensions, which may be of independent interest. Joint work with Boris Bukh.

Ander Lamaison, Uniform Turán density beyond 3-graphs
2024-02-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The uniform Turán density $pi_u(F)$ of a hypergraph $F$, introduced by Erdős and Sós, is the smallest value of $d$ such that any hypergraph $H$ where all linear-sized subsets of vertices of $H$ have density greater than $d$ contains $F$ as a subgraph. Over the past few years the value of $pi_u(F)$ was determined for several classes of 3-graphs, but no nonzero value of $pi_u(F)$ has been found for $r$-graphs with $r>3$. In this talk we show the existence of $r$-graphs $F$ with $pi_u(F)={r choose 2}^{-{r choose 2}}$, which we conjecture is minimum possible. Joint work with Frederik Garbe, Daniel Il’kovic, Dan Král’ and Filip Kučerák.

Sebastian Wiederrecht, Packing even directed circuits quarter-integrally
2024-02-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove the existence of a computable function $fcolonmathbb{N}tomathbb{N}$ such that for every integer $k$ and every digraph $D$ either contains a collection $mathcal{C}$ of $k$ directed cycles of even length such that no vertex of $D$ belongs to more than four cycles in $mathcal{C}$, or there exists a set $Ssubseteq V(D)$ of size at most $f(k)$ such that $D-S$ has no directed cycle of even length.
This is joint work with Maximilian Gorsky, Ken-ichi Kawarabayashi, and Stephan Kreutzer.

Jie Han (韩杰), Perfect matchings in dense uniform hypergraphs
2024-02-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
There has been a raising interest on the study of perfect matchings in uniform hypergraphs in the past two decades, including extremal problems and their algorithmic versions. I will introduce the problems and some recent developments.

William Cook, The Traveling Salesman Problem: Amazon Deliveries, Pub Walks, and Astro Tours
2024-03-06 16:00 - 17:00 | Room 1501, Bldg. E2-2, KAIST
Amazon drivers hit the road every day, each taking a Prime van in a traveling salesman problem tour through 150 or more customer stops. But even a whisper of the TSP strikes fear in the heart of the computing world. A Washington Post article reported it would take “1,000 years to compute the most efficient route between 22 cities.” Claims such as this, however, ignore 70 years of intense study. A 22-city TSP can be handled in a snap with modern methods, even on an iPhone. And, coming to the aid of Amazon drivers, we describe techniques used to win the $100,000 top prize in the Amazon Last Mile Routing Challenge, together with Stephan Held and Keld Helsgaun.
Going larger, we discuss methods used to find to precise optimality the shortest walking tour to 49,687 pubs in the UK. Even larger, if you want to visit 136,606,128 stars, there is a 3D route, ready to go, that is guaranteed to be no more than 1.00018 times longer that a shortest-possible tour.
The general setting is the following. Complexity theory suggests there are limits to the power of general-purpose computational techniques, in engineering, science and elsewhere. But what are these limits and how widely do they constrain our quest for knowledge? The TSP can play a crucial role in this discussion, demonstrating whether or not focused efforts on a single, possibly unsolvable, model will produce results beyond our expectations.

Linda Cook, On polynomial degree-boundedness
2024-03-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rzążewski, Thomassé, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$ contains either $K_{s,s}$ as a subgraph or contains an induced subdivision of $H$. This improves upon a result of Kühn and Osthus from 2004 who proved it for graphs whose average degree is at least triply exponential in $s$ and a recent result of Du, Girão, Hunter, McCarty and Scott for graphs with average degree at least singly exponential in $s$.
As an application, we prove that the class of graphs that do not contain an induced subdivision of $K_{s,t}$ is polynomially $chi$-bounded. In the case of $K_{2,3}$, this is the class of theta-free graphs, and answers a question of Davies. Along the way, we also answer a recent question of McCarty, by showing that if $mathcal{G}$ is a hereditary class of graphs for which there is a polynomial $p$ such that every bipartite $K_{s,s}$-free graph in $mathcal{G}$ has average degree at most $p(s)$, then more generally, there is a polynomial $p’$ such that every $K_{s,s}$-free graph in $mathcal{G}$ has average degree at most $p'(s)$. Our main new tool is an induced variant of the Kővári-Sós-Turán theorem, which we find to be of independent interest.
This is joint work with Romain Bourneuf (ENS de Lyon), Matija Bucić (Princeton), and James Davies (Cambridge),

Evangelos Protopapas, Erdős-Pósa Dualities for Minors
2024-03-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let $mathcal{G}$ and $mathcal{H}$ be minor-closed graphs classes. The class $mathcal{H}$ has the Erdős-Pósa property in $mathcal{G}$ if there is a function $f : mathbb{N} to mathbb{N}$ such that every graph $G$ in $mathcal{G}$ either contains (a packing of) $k$ disjoint copies of some subgraph minimal graph $H
otin mathcal{H}$ or contains (a covering of) $f(k)$ vertices, whose removal creates a graph in $mathcal{H}$. A class $mathcal{G}$ is a minimal EP-counterexample for $mathcal{H}$ if $mathcal{H}$ does not have the Erdős-Pósa property in $mathcal{G}$, however it does have this property for every minor-closed graph class that is properly contained in $mathcal{G}$. The set $frak{C}_{mathcal{H}}$ of the subset-minimal EP-counterexamples, for every $mathcal{H}$, can be seen as a way to consider all possible Erdős-Pósa dualities that can be proven for minor-closed classes. We prove that, for every $mathcal{H}$, $frak{C}_{mathcal{H}}$ is finite and we give a complete characterization of it. In particular, we prove that $|frak{C}_{mathcal{H}}| = 2^{operatorname{poly}(ell(h))}$, where $h$ is the maximum size of a minor-obstruction of $mathcal{H}$ and $ell(cdot)$ is the unique linkage function. As a corollary of this, we obtain a constructive proof of Thomas’ conjecture claiming that every minor-closed graph class has the half-integral Erdős-Pósa property in all graphs.
This is joint work with Christophe Paul, Dimitrios Thilikos, and Sebastian Wiederrecht.

Casey Tompkins, On graphs without cycles of length 0 modulo 4
2024-04-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Bollobás proved that for every $k$ and $ell$ such that $kmathbb{Z}+ell$ contains an even number, an $n$-vertex graph containing no cycle of length $ell bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycle of length $0 bmod 4$.
This is joint work with Ervin Győri, Binlong Li, Nika Salia, Kitti Varga and Manran Zhu.

Eero Räty, Positive discrepancy, MaxCut and eigenvalues of graphs
2024-04-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The positive discrepancy of a graph $G$ of edge density $p$ is defined as the maximum of $e(U) – p|U|(|U|-1)/2$, where the maximum is taken over subsets of vertices in G. In 1993 Alon proved that if G is a $d$-regular graph on $n$ vertices and $d = O(n^{1/9})$, then the positive discrepancy of $G$ is at least $c d^{1/2}n$ for some constant $c$.
We extend this result by proving lower bounds for the positive discrepancy with average degree d when $d < (1/2 – epsilon)n$. We prove that the same lower bound remains true when $d < n^(2/3)$, while in the ranges $n^{2/3} < d < n^{4/5}$ and $n^{4/5} < d < (1/2 – epsilon)n$ we prove that the positive discrepancy is at least $n^2/d$ and $d^{1/4}n/log(n)$ respectively.
Our proofs are based on semidefinite programming and linear algebraic techniques. Our results are tight when $d < n^{3/4}$, thus demonstrating a change in the behaviour around $d = n^{2/3}$ when a random graph no longer minimises the positive discrepancy. As a by-product, we also present lower bounds for the second largest eigenvalue of a $d$-regular graph when $d < (1/2 – epsilon)n$, thus extending the celebrated Alon-Boppana theorem.
This is joint work with Benjamin Sudakov and István Tomon.

Magnus Wahlström, Algorithmic aspects of linear delta-matroids
2024-04-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Delta-matroids are a generalization of matroids with connections to many parts of graph theory and combinatorics (such as matching theory and the structure of topological graph embeddings). Formally, a delta-matroid is a pair $D=(V,mathcal F)$ where $mathcal F$ is a collection of subsets of V known as “feasible sets.” (They can be thought of as generalizing the set of bases of a matroid, while relaxing the condition that all bases must have the same cardinality.)
Like with matroids, an important class of delta-matroids are linear delta-matroids, where the feasible sets are represented via a skew-symmetric matrix. Prominent examples of linear delta-matroids include linear matroids and matching delta-matroids (where the latter are represented via the famous Tutte matrix).
However, the study of algorithms over delta-matroids seems to have been much less developed than over matroids.
In this talk, we review recent results on representations of and algorithms over linear delta-matroids. We first focus on classical polynomial-time aspects. We present a new (equivalent) representation of linear delta-matroids that is more suitable for algorithmic purposes, and we show that so-called delta-sums and unions of linear delta-matroids are linear. As a result, we get faster (randomized) algorithms for Linear Delta-matroid Parity and Linear Delta-matroid Intersection, improving results from Geelen et al. (2004).
We then move on to parameterized complexity aspects of linear delta-matroids. We find that many results regarding linear matroids which have had applications in FPT algorithms and kernelization directly generalize to linear delta-matroids of bounded rank. On the other hand, unlike with matroids, there is a significant difference between the “rank” and “cardinality” parameters – the structure of bounded-cardinality feasible sets in a delta-matroid of unbounded rank is significantly harder to deal with than feasible sets in a bounded-rank delta-matroid.

Víctor Dalmau, Right-adjoints of Datalog Programs
2024-04-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures A and B, we have that Λ(A) maps homomorphically to B if and only if A maps homomorphically to Γ(B). If this is the case Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We shall present several recent advances in this direction including a new approach based on the notion of Datalog Program borrowed from logic.

Maximilian Gorsky, Towards the half-integral Erdős-Pósa property for even dicycles
2024-04-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A family $mathcal F$ of (di)graphs is said to have the half- or quarter-integral Erdős-Pósa property if, for any integer $k$ and any (di)graph $G$, there either exist $k$ copies of graphs in $mathcal F$ within $G$ such that any vertex of $G$ is contained in at most 2, respectively at most 4, of these copies, or there exists a vertex set $A$ of size at most $f(k)$ such that $G – A$ contains no copies of graphs in $mathcal F$. Very recently we showed that even dicycles have the quarter-integral Erdős-Pósa property [STOC’24] via the proof of a structure theorem for digraphs without large packings of even dicycles.
In this talk we discuss our current effort to improve this approach towards the half-integral Erdős-Pósa property, which would be best possible, as even dicycles do not have the integral Erdős-Pósa property. Complementing the talk given by Sebastian Wiederrecht in this seminar regarding our initial result, we also shine a light on some of the particulars of the embedding we use in lieu of flatness and how this helps us to move even dicycles through the digraph. In the process of this, we highlight the parts of the proof that initially caused the result to be quarter-integral.
(This is joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and Sebastian Wiederrecht.)

Tony Huynh, Aharoni's rainbow cycle conjecture holds up to an additive constant
2024-05-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture for digraphs. If G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉.
In this talk, we prove that Aharoni’s conjecture holds up to an additive constant. Specifically, we show that for each fixed r, there exists a constant c such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most n/r+c.
This is joint work with Patrick Hompe.

Niloufar Fuladi, Cross-cap drawings and signed reversal distance
2024-05-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A cross-cap drawing of a graph G is a drawing on the sphere with g distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph G with g cross-caps can be used to represent an embedding of G on a non-orientable surface of genus g. Mohar conjectured that any triangulation of a non-orientable surface of genus g admits a cross-cap drawing with g cross-caps in which each edge of the triangulation enters each cross-cap at most once. Motivated by Mohar’s conjecture, Schaefer and Stefankovic provided an algorithm that computes a cross-cap drawing with a minimal number of cross-caps for a graph G such that each edge of the graph enters each cross-cap at most twice. In this talk, I will first outline a connection between cross-cap drawings and an algorithm coming from computational biology to compute the signed reversal distance between two permutations. This connection will then be leveraged to answer two computational problems on graphs embedded on surfaces.
First, I show how to compute a “short” canonical decomposition for a non-orientable surface with a graph embedded on it. Such canonical decompositions were known for orientable surfaces, but the techniques used to compute them do not generalize to non-orientable surfaces due to their more complex nature. Second, I explain how to build a counter example to a stronger version of Mohar’s conjecture that is stated for pseudo-triangulations.
This is joint work with Alfredo Hubard and Arnaud de Mesmay.

Vadim Lozin, Graph problems and monotone classes
2024-05-21 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem on monotone classes.

Yongho Shin (신용호), Three-way online correlated selection
2024-05-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm’s choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break a natural long-standing barrier in edge-weighted online bipartite matching. They posed two open questions, one of which was the following: Can we obtain n-way OCS for $n >2$, in which the algorithm can be given $n >2$ elements to choose from at each timestep?
In this talk, we affirmatively answer this open question by presenting a three-way OCS which is simple to describe: it internally runs two instances of two-way OCS, one of which is fed with the output of the other. Contrast to its simple construction, we face a new challenge in analysis that the final output probability distribution of our three-way OCS is highly elusive since it requires the actual output distribution of two-way OCS. We show how we tackle this challenge by approximating the output distribution of two-way OCS by a flatter distribution serving as a safe surrogate.
This is joint work with Hyung-Chan An.

Jane Tan, Semi-strong colourings of hypergraphs
2024-06-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $min{c, |e|}$ distinct colours. Let $chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $chi(t,c)$ is finite: they showed that $chi(t,c)$ is finite whenever $t geq c$ and unbounded when $tleq c-2$. The boundary case $chi(c-1, c)$ has remained elusive for some time: $chi(1,2)$ is known to be finite by an easy classical result, and $chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gyárfás in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kamčev in which we fill in this gap by showing that $chi(c-1, c)$ is finite in general.

Maria Chudnovsky, Anticomplete subgraphs of large treewidth
2024-06-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will discuss recent progress on the topic of induced subgraphs and tree-decompositions. In particular this talk with focus on the proof of a conjecture of Hajebi that asserts that (if we exclude a few obvious counterexamples) for every integer t, every graph with large enough treewidth contains two anticomplete induced subgraphs each of treewidth at least t. This is joint work with Sepher Hajebi and Sophie Spirkl.
