Discrete Mathematics Group

Magnus Wahlström, Algorithmic aspects of linear delta-matroids
2024-04-16 16:30 - 17:30 (Asia/Seoul) | 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.

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.

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.

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.

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),

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

Robert Hickingbotham, Powers of planar graphs, product structure, and blocking partitions
2023-10-24 16:30 - 17:30 | 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.

Matija Bucić, Essentially tight bounds for rainbow cycles in proper edge-colourings
2023-10-17 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon’s bound. We show that the answer to the question is equal to $(log n)^{1+o(1)}$.
A key tool we use is the theory of robust sublinear expanders. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.
Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.

2023 Vertex-Minor Workshop
2023-10-15 - 2023-10-20 | SONO BELLE Jeju
This workshop aims to foster collaborative discussions and explore the various aspects of vertex-minors, including structural theory and their applications.
This event will be small-scale, allowing for focused talks and meaningful interactions among participants.
Website: https://indico.ibs.re.kr/event/596/

Domagoj Bradač, Effective bounds for induced size-Ramsey numbers of cycles
2023-10-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The k-color induced size-Ramsey number of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles these numbers are linear for any constant number of colours, i.e., for some C=C(k), there is a graph with at most Cn edges whose any k-edge-coloring contains a monochromatic induced cycle of length n. The value of C comes from the use of the sparse regularity lemma and has a tower-type dependence on k. In this work, we obtain nearly optimal bounds for the required value of C. Joint work with Nemanja Draganić and Benny Sudakov.

Carl R. Yerger, Solving Problems in Graph Pebbling using Optimization and Structural Techniques
2023-09-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebbling on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified target vertex.
In this talk, we will give a survey of several streams of research in pebbling, including describing a theoretical and computational framework that uses mixed-integer linear programming to obtain bounds for the pebbling numbers of graphs. We will also discuss improvements to this framework through the use of newly proved weight functions that strengthen the weight function technique of Hurlbert. Finally, we will discuss some open extremal problems in pebbling, specifically related to Class 0 graphs and describe how structural graph theoretic techniques such as discharging can be used to obtain results.
Collaborators on these projects include Dan Cranson, Dominic Flocco, Luke Postle, Jonad Pulaj, Chenxiao Xue, Marshall Yang, Daniel Zhou.

Donggyu Kim (김동규), Orthogonal matroids over tracts
2023-09-19 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Even delta-matroids generalize matroids, as they are defined by a certain basis exchange axiom weaker than that of matroids. One natural example of even delta-matroids comes from a skew-symmetric matrix over a given field $K$, and we say such an even delta-matroid is representable over the field $K$. Interestingly, a matroid is representable over $K$ in the usual manner if and only if it is representable over $K$ in the sense of even delta-matroids. The representability of matroids got much interest and was extensively studied such as excluded minors and representability over more than one field. Recently, Baker and Bowler (2019) integrated the notions of $K$-representable matroids, oriented matroids, and valuated matroids using tracts that are commutative ring-like structures in which the sum of two elements may output no element or two or more elements.
We generalize Baker-Bowler’s theory of matroids with coefficients in tracts to orthogonal matroids that are equivalent to even delta-matroids. We define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets, and orthogonal vector sets, and establish basic properties on functoriality, duality, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular (i.e., representable over all fields) if and only if it is representable over $mathbb{F}_2$ and $mathbb{F}_3$, which was originally shown by Geelen (1996), and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over $mathbb{F}_3$ and $mathbb{F}_4$.
This is joint work with Tong Jin.

Seog-Jin Kim (김석진), The square of every subcubic planar graph of girth at least 6 is 7-choosable
2023-09-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Wegner’s conjecture (1977) states that for a planar graph $G$, the chromatic number $chi(G^2)$ of $G^2$ is at most 7 if $Delta(G) = 3$, at most $Delta(G)+5$ if $4 leq Delta(G) leq 7$, and at most $lfloor frac{3 Delta(G)}{2} rfloor$ if $Delta(G) geq 8$. Wegner’s conjecture is still wide open. The only case for which we know tight bound is when $Delta(G) = 3$. Thomassen (2018) showed that $chi(G^2) leq 7$ if $G$ is a planar graph with $Delta(G) = 3$, which implies that Wegner’s conjecture is true for $Delta(G) = 3$. A natural question is whether $chi_{ell}(G^2) leq 7$ or not if $G$ is a subcubic planar graph, where $chi_{ell}(G^2)$ is the list chromatic number of $G^2$. Cranston and Kim (2008) showed that $chi_{ell}(G^2) leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $chi_{ell}(G^2) leq 7$ if $G$ is a subcubic planar graph of girth at least 6. This is joint work with Xiaopan Lian (Nankai University).

Sebastian Wiederrecht, Delineating half-integrality of the Erdős-Pósa property for minors
2023-09-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 1986, Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minor if and only if it is planar. In particular, for every non-planar graph $H$ they gave examples showing that the Erdős-Pósa property does not hold for $H$. Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors.
In this talk, we start the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph $H$ there exists a finite family $mathfrak{F}_H$ of parametric graphs such that $H$ has the Erdős-Pósa property in a minor-closed graph class $mathcal{G}$ if and only if $mathcal{G}$ excludes a minor of each of the parametric graphs in $mathfrak{F}_H$. We prove this conjecture for the class $mathcal{H}$ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar $Hinmathcal{H}$ the family $mathfrak{F}_H$ can be chosen to be precisely the two families of Robertson-Seymour counterexamples to the Erdős-Pósa property of $H$.

Dabeen Lee (이다빈), From coordinate subspaces over finite fields to ideal multipartite uniform clutters
2023-08-29 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Take a prime power $q$, an integer $ngeq 2$, and a coordinate subspace $Ssubseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $mathcal{C}$. In this paper, we determine when the clutter $mathcal{C}$ is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $tau=2$ conjectures for this class of clutters. This is joint work with Ahmad Abdi (London School of Economics).
