Tomáš Masařík, Separator Theorem for Minor-free Graphs in Linear Time
2026-01-20 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
The planar separator theorem by Lipton and Tarjan [FOCS ’77, SIAM Journal on Applied Mathematics ’79] states that any planar graph with $n$ vertices has a balanced separator of size $O(sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan’s theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC ’90, Journal of the AMS ’90] showed that any minor-free graph admits a balanced separator of size $O(sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(sqrt{n})$ or have superlinear running time, or both.
In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.
This is a joint work with Édouard Bonnet, Tuukka Korhonen, Hung Le, and Jason Li.

Daniel Dadush, A Strongly Polynomial Algorithm for Linear Programs with at Most Two Non-Zero Entries per Row or Column
2026-01-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh (MOR ’17) and Megiddo (SICOMP ’83) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row
Joint work with Zhuan Khye Koh (Boston U), Bento Natura (Columbia), Neil Olver (LSE), and László A. Végh (Bonn).

Xiaofan Yuan (袁晓璠), Rainbow structures in edge colored graphs
2026-02-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Let $G = (V, E)$ be a graph on $n$ vertices, and let $c : E to P$, where $P$ is a set of colors. Let $delta^c(G) = min_{v in V} { d^{c}(v) }$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $delta^c(G)geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$, the same bound for $delta^c(G)$ implies that any two vertices are connected by a rainbow path. We also show sufficient conditions on $delta^c(G)$ for the existence of a rainbow cycle of length $2k$ in sufficiently large bipartite graphs $G$, which are tight in many cases. This is joint work with Andrzej Czygrinow.

Seonghun Park (박성훈), Formalizing Flag Algebras in the Lean Theorem Prover
2026-02-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Flag algebras are a mathematical framework introduced by Alexander Razborov in 2007, which has been used to resolve a wide range of open problems in extremal graph theory in the past twenty years. This framework provides an algebraic setup where one can express relationships between induced subgraph densities symbolically. It also comes with mathematical techniques for systematically deriving such relationships that always hold. Some of these techniques have even been implemented in automatic tools, such as Flagmatic. In this work, we formalise flag algebras in Lean, an interactive theorem prover based on dependent type theory. Lean is computer software that lets us write mathematical definitions and proofs in a computer and check the correctness of written proofs using a computer. By formalizing flag algebras in Lean, we can ensure that the mathematical foundation of flag algebras does not have any mistakes, and provide a mathematical guarantee that theorems proved by flag algebras are indeed correct. In this talk, I will explain flag algebras and our Lean formalization using Mantel’s theorem as a guiding example. In the process, I will describe the challenges and lessons learned during the formalization.
This is a joint work with Jihoon Hyun, Gyeongwon Jeong, Sang-il Oum, and Hongseok Yang.

Marek Sokołowski, Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP
2026-02-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In this talk, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within $widetilde{O}(m+n^{2-varepsilon})$ work and $widetilde{O}(n^{1-varepsilon})$ depth for some positive $varepsilon>0$. For dense graphs with non-negative real weights, this yields the first nearly work-efficient strongly polynomial algorithm with sublinear depth. Moreover, we develop efficient parallel algorithms in the Word RAM model for several variants of SSSP in graphs with exponentially large edge weights.
This is a joint work with Adam Karczmarz and Wojciech Nadara.

Chính T. Hoàng, Problems on graph coloring
2026-03-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A k-coloring of a graph is an assignment of k colors to its vertices such that no two adjacent adjacent vertices receive the same color. The Coloring Problem is the problem of determining the smallest k such that the graph admits a k-coloring. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. The complexity of the Coloring Problem for L-free graphs is known (NP-complete or polynomial-time solvable) whenever L contains a single graph. There has been keen interest in coloring graphs whose forbidden list L contains basic graphs such as induced paths, induced cycles and their complements. In this talk, I will provide a survey of recent progress on this topic.

Dario Cavallaro, Well-quasi-ordering Eulerian directed graphs by (strong) immersion
2026-03-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Directed graphs prove to be very hard to tame in contrast to undirected graphs. In particular, they are not well-quasi-ordered by any known relevant inclusion relation, and are lacking fruitful structure theorems. This motivates the search for structurally rich subclasses of directed graphs that are well behaved. Eulerian directed graphs are a particularly prominent example, sharing many similarities with undirected graphs. In fact, it is conjectured that Eulerian directed graphs are well-quasi-ordered by weak immersion, and even well-quasi-ordered by strong immersion when restricting to classes of bounded degree. We believe that we have a proof of both conjectures, and I will report on the current status, progress, and steps towards said proof and its implications. This is joint work with Ken-ichi Kawarabayashi and Stephan Kreutzer.

József Balogh, Clique covers and decompositions of cliques of graphs
2026-03-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Two related papers will be discussed:
1. In 1966, Erdős, Goodman, and Pósa showed that if $G$ is an $n$-vertex graph, then at most $lfloor n^2/4 rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced complete bipartite graph. This was generalized independently by Győri–Kostochka, Kahn, and Chung, who showed that every $n$-vertex graph admits an edge-decomposition into cliques of total `cost’ at most $2 lfloor n^2/4 rfloor$, where an $i$-vertex clique has cost $i$. Erdős suggested the following strengthening: every $n$-vertex graph admits an edge-decomposition into cliques of total cost at most $lfloor n^2/4 rfloor$, where now an $i$-vertex clique has cost $i-1$. We prove fractional relaxations and asymptotically optimal versions of both this conjecture and a conjecture of Dau, Milenkovic, and Puleo on covering the $t$-vertex cliques of a graph instead of the edges. Our proofs introduce a general framework for these problems using Zykov symmetrization, the Frankl–Rödl nibble method, and the Szemerédi Regularity Lemma. It is joint work with Jialin He, Robert Krueger, The Nguyen, and Michael Wigal.
2. Let $r ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Győri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Turán graph on $n$ vertices and $r – 1$ parts, then $G$ has at least $(2 – o(1))k/r$ edge-disjoint $r$-cliques. We prove this conjecture. It is joint work with Michael Wigal.

Hidde Koerts, Characterizing large clique number in tournaments
2026-03-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A backedge graph of a tournament $T$ with respect to a total ordering $prec$ of the vertices of $T$ is a graph on $V(T)$ where $uv$ is an edge if and only if $uv in A(T)$ and $v prec u$. In 2023, Aboulker, Aubian, Charbit and Lopes introduced the clique number of tournaments based on backedge graphs as a natural counterpart to the dichromatic number of tournaments. Specifically, the clique number of a tournament is the minimum clique number of a backedge graph when considering all possible orderings.
Given this definition, it is not immediately clear what the canonical clique object should be. In this talk, we provide an answer to this question. We show that if a tournament has large clique number, it contains a reasonably large subtournament from one of two simple and previously studied families of tournaments of unbounded clique number.
This talk is based on joint work with Logan Crew, Xinyue Fan, Benjamin Moore, and Sophie Spirkl.

Tung H. Nguyen, Polynomial χ-boundedness for excluding the five-vertex path
2026-03-31 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We overview the recent resolution of a 1985 open problem of Gyárfás, that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path. The proof introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erdős–Hajnal result for the five-vertex path.

Mamadou Moustapha Kanté, Strongly flip-flat classes of graphs
2026-04-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. Almost-wideness is a notion that was central in different characterisations of nowhere dense classes of graphs, and in particular the game-theoretic one. In this talk I will present the flip-flatness notions and conjectures about the characterization of strongly flip-flat graph classes. Then, I will present a proof that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide, making a step towards their characterisation. A consequence is a characterization of strongly flip-flat graph classes with low rank-depth colourings.
This is a joint work with F. Ghasemi, J. Grange and F. Madelaine.

Xin Wei, Separating hash families with large universe
2026-04-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Separating hash families are useful combinatorial structures that generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of the universe for a $t$-perfect hash family of length $N$ over an alphabet of size $q$. We show that $q^{2 – o(1)} < p_t(t, q) = o(q^2)$ for all $t ge 3$, thereby resolving an open problem raised by Blackburn et al. (2008) for certain parameter ranges. Previously, this result was known only for $t = 3$ and $t = 4$. Our approach establishes the existence of a large set of integers that avoids nontrivial solutions to a system of correlated linear equations. This is joint work with Xiande Zhang and Gennian Ge.

Maximilian Gorsky, The Disjoint Paths Problem lies in the Oort cloud of algorithms
2026-05-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In this talk we discuss recent work to that establishes that the bounds of the Vital Linkage Function is single-exponential. This has immediate impacts on the complexity of the k-Disjoint Paths Problem, Minor Checking, and more generally, the Folio-Problem. We in fact prove something even stronger: It turns out that it is not in fact the number of terminals (or more generally vertices) that matters in these problems, but rather their structure within the graph. Concretely, we show that the Vital Linkage Function is single-exponential only in the bidimensionality of the terminals, whilst the number of terminals contributes only polynomially. A direct consequence of this is an algorithm for the k-Disjoint Paths Problem running in $f(k)n^2$-time, where f(k) is singly exponential in k and doubly exponential in the bidimensionality of k. This derives directly from an algorithm for the Folio-problem we give that has an analogous runtime. Notably these are the first algorithms for these problems in which the function f is explicit. In particular, we give the first explicit bounds for the Vital Linkage Function.
Joint work with Dario Cavallaro, Stephan Kreutzer, Dimitrios Thilikos, and Sebastian Wiederrecht.

Benjamin Duhamel, Excluding a forest induced minor
2026-05-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We give an induced counterpart of the Forest Minor theorem: for any t ≥ 2, the $K_{t,t}$-subgraph-free H-induced-minor-free graphs have bounded pathwidth if and only if H belongs to a class F of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs H for which every weakly sparse H-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.
This is a joint work with É. Bonnet and R. Hickingbotham.

Xavier Goaoc, TBA
2026-05-19 16:30 - 17:30 | Room B332, IBS (기초과학연구원)

Sarah Morell, Unsplittable Transshipments
2026-05-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We consider an arc-capacitated directed graph $D=(V,A)$, where each node $v$ is associated with a rational balance value $b(v)$. Nodes with negative balance values are referred to as sources, while those with positive balance values are called sinks. A feasible $b$-transshipment is a flow $f : A to mathbb{R}_{ge 0}$ that routes the total supply of the sources to the sinks through $D$, while respecting the given arc capacity constraints and satisfying the balance requirements at each node. An unsplittable $b$-transshipment additionally requires that, for each source-sink pair, the flow sent from that source to that sink is routed along at most one directed path. Unsplittable $b$-transshipments (UT) generalize the well-studied single source unsplittable flow (SSUF) problem in which $D$ contains a single source and multiple sinks, and each demand must be routed along a single path from the common source to its destination.
Given a feasible $b$-transshipment $f$ that is not necessarily unsplittable, a natural question is whether there exists an feasible unsplittable $b$-transshipment flow $g$ that approximates $f$ in an arc-wise sense. In particular, we seek bounds on the maximum deviation $|f_a-g_a|$ over all arcs $a in A$. For the special case of SSUFs, Dinitz, Garg, and Goemans (Combinatorica 1999) proved that there exists an unsplittable flow $g$ such that $g_a leq f_a + d_{max}$ for all $a in A$, where $d_{max}$ denotes the maximum demand value. Jointly with Martin Skutella (Mathematical Programming 2022), we studied unsplittable flows with arc-wise lower bounds and showed that there exists an unsplittable flow $g$ satisfying $g_a ge f_a – d_{max}$ for all $a in A$.
In this talk, we extend this line of research by adapting the techniques of Dinitz, Garg, and Goemans to the more general setting of UTs. We show that, given any feasible $b$-transshipment $f$, there exists a feasible unsplittable $b$-transshipment $g$ such that $g_a leq f_a + d_{max}$ (resp. $g_a ge f_a – d_{max}$) for all $a in A$.
This is joint work with Srinwanti Debgupta and Martin Skutella.

Maria Chudnovsky, TBA
2026-06-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)

Martin Milanič, TBA
2026-06-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)

2026 Workshop on Topological Combinatorics
2026-06-28 - 2026-07-11 | GIST
The 2026 Workshop on Topological Combinatorics will be held from June 28 to July 11, 2026 at Gwangju Institute of Science and Technology (GIST), located in Gwangju in the southwest of Republic of Korea.
The workshop aims to bring together researchers interested in applications of topology to combinatorics and related areas. This will be the fifth workshop in a series initiated by Ron Aharoni (August 2018 in Shantou, July 2019 in Prague, July 2020 on Zoom, and June 2024 in Paris).
The first week (June 28 – July 04) will be reserved for presentations, while the following days (July 05 – July 11) will be for discussion and collaborations.
The campus offers a variety of amenities, including a guesthouse that can accommodate approximately 20 guests.
In addition, various lodging options are available within a 15–20 minute walking distance.
We are planning to provide accommodation for all invited participants, and we are also considering offering partial travel support for those with limited funding.

Website: https://sites.google.com/view/2026topocomb

Ting-Wei Chao, Entropy method and mixture bound
2026-07-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The entropy method has been used in many recent works in extremal combinatorics. With the help of Shannon entropy, significant progress has been made on several classical problems, such as the union-closed conjecture and the Sidorenko conjecture. In our recent work, we use the entropy method to give new proofs of the Kruskal–Katona theorem and Turán’s theorem, as well as some of their generalizations. The new ingredient in our approach is a method for upper bounding the sum of $2^{mathbb{H}(X_i)}$ for random variables $X_1,cdots,X_k$ whose supports do not overlap too much. We call this method the mixture bound, and it can be viewed as an entropic version of double counting. In this talk, I will introduce the mixture bound and show some examples of how it can be applied on colorful versions of the Kruskal–Katona theorem. Base on joint work with Maya Sankar and Hung-Hsun Hans Yu.

2026 Korean Student Combinatorics Workshop
2026-07-27 - 2026-07-31 | Gongju Hanok Village, Gongju
Website: https://kscw.combinatorics.kr/

Jinyoung Park (박진영), TBA
2026-08-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
