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PRODID:-//Discrete Mathematics Group - ECPv6.15.20//NONSGML v1.0//EN
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X-WR-CALNAME:Discrete Mathematics Group
X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
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X-Robots-Tag:noindex
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BEGIN:VTIMEZONE
TZID:Asia/Seoul
BEGIN:STANDARD
TZOFFSETFROM:+0900
TZOFFSETTO:+0900
TZNAME:KST
DTSTART:20210101T000000
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230809T163000
DTEND;TZID=Asia/Seoul:20230809T173000
DTSTAMP:20260506T071212
CREATED:20230403T234729Z
LAST-MODIFIED:20240705T164149Z
UID:6963-1691598600-1691602200@dimag.ibs.re.kr
SUMMARY:R. Amzi Jeffs\, Intersection patterns of convex sets
DESCRIPTION:How can one arrange a collection of convex sets in d-dimensional Euclidean space? This guiding question is fundamental in discrete geometry\, and can be made concrete in a variety of ways\, for example the study of hyperplane arrangements\, embeddability of simplicial complexes\, Helly-type theorems\, and more. This talk will focus on the classical topic of d-representable complexes and its more recent generalization to convex codes. We will show how these frameworks differ\, share some novel Helly-type results\, and offer several tantalizing open questions.
URL:https://dimag.ibs.re.kr/event/2023-08-09/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230802T163000
DTEND;TZID=Asia/Seoul:20230802T173000
DTSTAMP:20260506T071212
CREATED:20230506T225557Z
LAST-MODIFIED:20240705T163040Z
UID:7156-1690993800-1690997400@dimag.ibs.re.kr
SUMMARY:Daniel Kráľ\, High chromatic common graphs
DESCRIPTION:Ramsey’s Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. As probabilistic constructions often provide good bounds on quantities in extremal combinatorics\, we say that a graph H is common if the random 2-edge-coloring asymptotically minimizes the number of monochromatic copies of H. This notion goes back to the work of Erdős in the 1960s\, who conjectured that every complete graph is common. The conjecture was disproved by Thomason in the 1980s\, however\, a classification of common graphs remains one of the most intriguing problems in extremal combinatorics. \nSidorenko’s Conjecture (if true) would imply that every bipartite graph is common\, and in fact\, no bipartite common graph unsettled for Sidorenko’s Conjecture is known. Until Hatami et al. showed that a 5-wheel is common about a decade ago\, all graphs known to be common had chromatic number at most three. The existence of a common graph with chromatic number five or more has remained open for three decades. \nWe will present a construction of (connected) common graphs with arbitrarily large chromatic number. At the end of the talk\, we will also briefly discuss the extension of the notion to more colors and particularly its relation to Sidorenko’s Conjecture. \nThe main result presented in the talk is based on joint work with Jan Volec and Fan Wei.
URL:https://dimag.ibs.re.kr/event/2023-08-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230725T163000
DTEND;TZID=Asia/Seoul:20230725T173000
DTSTAMP:20260506T071212
CREATED:20230615T122924Z
LAST-MODIFIED:20240707T073405Z
UID:7282-1690302600-1690306200@dimag.ibs.re.kr
SUMMARY:Dong Yeap Kang (강동엽)\, Hamilton cycles and optimal matchings in a random subgraph of uniform Dirac hypergraphs
DESCRIPTION:A loose cycle is a cyclic ordering of edges such that every two consecutive edges share exactly one vertex. A cycle is Hamilton if it spans all vertices. A codegree of a $k$-uniform hypergraph is the minimum nonnegative integer $t$ such that every subset of vertices of size $k-1$ is contained in $t$ distinct edges. \nWe prove “robust” versions of Dirac-type theorems for Hamilton cycles and optimal matchings. \nLet $\mathcal{H}$ be a $k$-uniform hypergraph on $n$ vertices with $n \in (k-1)\mathbb{N}$ and codegree at least $n/(2k-2)$\, and let $\mathcal{H}_p$ be a spanning subgraph of $\mathcal{H}$ such that each edge of $\mathcal{H}$ is included in $\mathcal{H}_p$ with probability $p$ independently at random. We prove that a.a.s. $\mathcal{H}_p$ contains a loose Hamilton cycle if $p = \Omega(n^{-k+1} \log n)$\, which is asymptotically best possible. We also present similar results for Hamilton $\ell$-cycles for $\ell \geq 2$. \nFurthermore\, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph on $n$ vertices with $n \notin k \mathbb{N}$ and codegree at least $\lfloor n/k \rfloor$\, then a.a.s. $\mathcal{H}_p$ contains a matching of size $\lfloor n/k \rfloor$ if $p = \Omega(n^{-k+1} \log n)$. This is also asymptotically best possible. \nThis is joint work with Michael Anastos\, Debsoumya Chakraborti\, Abhishek Methuku\, and Vincent Pfenninger.
URL:https://dimag.ibs.re.kr/event/2023-07-25/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230718T163000
DTEND;TZID=Asia/Seoul:20230718T173000
DTSTAMP:20260506T071212
CREATED:20230601T142456Z
LAST-MODIFIED:20240705T163017Z
UID:7226-1689697800-1689701400@dimag.ibs.re.kr
SUMMARY:Andrzej Grzesik\, Rainbow Turán problems
DESCRIPTION:In a rainbow variant of the Turán problem\, we consider $k$ graphs on the same set of vertices and want to determine the smallest possible number of edges in each graph\, which guarantees the existence of a copy of a given graph $F$ containing at most one edge from each graph. In other words\, we treat each of the $k$ graphs as a graph in one of the $k$ colors and consider how many edges in each color force a rainbow copy of a given graph $F$. In the talk\, we will describe known results on the topic\, as well as present recent developments\, obtained jointly with Sebastian Babiński and Magdalen Prorok\, solving the rainbow Turán problem for a path on 4 vertices and a directed triangle with any number of colors.
URL:https://dimag.ibs.re.kr/event/2023-07-18/
LOCATION:Room S221\, IBS (기초과학연구원) Science Culture Center
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230710T163000
DTEND;TZID=Asia/Seoul:20230710T173000
DTSTAMP:20260506T071212
CREATED:20230410T004309Z
LAST-MODIFIED:20240707T073602Z
UID:6987-1689006600-1689010200@dimag.ibs.re.kr
SUMMARY:Xuding Zhu (朱緒鼎)\, List version of 1-2-3 conjecture
DESCRIPTION:The well-known 1-2-3 Conjecture by Karoński\, Łuczak and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1\, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct. The list version of the 1-2-3 Conjecture by Bartnicki\, Grytczuk and Niwczyk states that the same holds if each edge $e$ has the choice of weights not necessarily from $\{1\,2\,3\}$\, but from any set $\{x(e)\,y(e)\,z(e)\}$ of three real numbers. The goal of this talk is to survey developments on the 1-2-3 Conjecture\, especially on the list version of the 1-2-3 Conjecture.
URL:https://dimag.ibs.re.kr/event/2023-07-10/
LOCATION:Room B109\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230704T163000
DTEND;TZID=Asia/Seoul:20230704T173000
DTSTAMP:20260506T071212
CREATED:20230627T045021Z
LAST-MODIFIED:20240705T162129Z
UID:7316-1688488200-1688491800@dimag.ibs.re.kr
SUMMARY:Tuan Tran\, Complexity of null dynamical systems
DESCRIPTION:A theoretical dynamical system is a pair (X\,T) where X is a compact metric space and T is a self homeomorphism of X. The topological entropy of a theoretical dynamical system (X\,T)\, first introduced in 1965 by Adler\, Konheim and McAndrew\, is a nonnegative real number that measures the complexity of the system. Systems with positive entropy are random in certain sense\, and systems with zero entropy are said to be deterministic. To distinguish between deterministic systems\, Huang and Ye (2009) introduced the concept of maximal pattern entropy of a theoretical dynamical system. At the heart of their argument is a Sauer-Shelah-type lemma. We will discuss this lemma and its surprising connection to a recent breakthrough in communication complexity. \nJoint work with Guorong Gao\, Jie Ma\, and Mingyuan Rong.
URL:https://dimag.ibs.re.kr/event/2023-07-04/
LOCATION:Room B109\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230627T163000
DTEND;TZID=Asia/Seoul:20230627T173000
DTSTAMP:20260506T071212
CREATED:20230601T063015Z
LAST-MODIFIED:20240707T073615Z
UID:7223-1687883400-1687887000@dimag.ibs.re.kr
SUMMARY:Chong Shangguan (上官冲)\, The hat guessing number of graphs
DESCRIPTION:Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph\, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors\, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously\, according to a predetermined guessing strategy and the hat colors they see\, where no communication between them is allowed. Given a graph $G$\, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. \nIn 2008\, Butler\, Hajiaghayi\, Kleinberg\, and Leighton asked whether the hat guessing number of the complete bipartite graph $K_{n\,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively\, showing that for sufficiently large $n$\, $HG(K_{n\,n})\ge n^{0.5-o(1)}$. Our guessing strategy is based on some ideas from coding theory and probabilistic method. \nBased on a joint work with Noga Alon\, Omri Ben-Eliezer\, and Itzhak Tamo.
URL:https://dimag.ibs.re.kr/event/2023-06-27/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230620T163000
DTEND;TZID=Asia/Seoul:20230620T173000
DTSTAMP:20260506T071212
CREATED:20230410T120415Z
LAST-MODIFIED:20240705T164137Z
UID:6995-1687278600-1687282200@dimag.ibs.re.kr
SUMMARY:Guanghui Wang (王光辉)\, Embeddings in uniformly dense  hypergraphs
DESCRIPTION:An archetype problem in extremal combinatorics is to study the structure of subgraphs appearing in different classes of (hyper)graphs. We will focus on such embedding problems in uniformly dense hypergraphs. In precise\, we will mention the uniform Turan density of some hypergraphs.
URL:https://dimag.ibs.re.kr/event/2023-06-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230613T163000
DTEND;TZID=Asia/Seoul:20230613T173000
DTSTAMP:20260506T071212
CREATED:20230410T043558Z
LAST-MODIFIED:20240707T073632Z
UID:6992-1686673800-1686677400@dimag.ibs.re.kr
SUMMARY:Minho Cho (조민호)\, Strong Erdős-Hajnal property on chordal graphs and its variants
DESCRIPTION:A graph class $\mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$\, either $G$ or its complement has $K_{m\, m}$ as a subgraph where $m \geq \left\lfloor c|V(G)| \right\rfloor$. We prove that the class of chordal graphs satisfies SEH-property with constant $c = 2/9$. \nOn the other hand\, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results\, we show that for every pair $F_1\, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves\, there exist subfamilies $F’_1 \subseteq F_1$ and $F’_2 \subseteq F_2$ of size $\theta \left( \frac{\ln k}{k} \left| F_1 \right|\right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal. \nJoint work with Andreas Holmsen\, Jinha Kim and Minki Kim.
URL:https://dimag.ibs.re.kr/event/2023-06-13/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230530T163000
DTEND;TZID=Asia/Seoul:20230530T173000
DTSTAMP:20260506T071212
CREATED:20230323T234815Z
LAST-MODIFIED:20240707T073642Z
UID:6946-1685464200-1685467800@dimag.ibs.re.kr
SUMMARY:Suyun Jiang (江素云)\, How connectivity affects the extremal number of trees
DESCRIPTION:The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. Despite significant interest\, the conjecture remains unsolved. Recently\, Caro\, Patkós\, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them\, for a $k$-vertex tree $T$\, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges\, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore\, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.
URL:https://dimag.ibs.re.kr/event/2023-05-30/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230516T163000
DTEND;TZID=Asia/Seoul:20230516T173000
DTSTAMP:20260506T071212
CREATED:20230116T010528Z
LAST-MODIFIED:20240707T073700Z
UID:6669-1684254600-1684258200@dimag.ibs.re.kr
SUMMARY:Oliver Janzer\, Small subgraphs with large average degree
DESCRIPTION:We study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$\, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor\, and resolves a conjecture of Feige and Wagner. In addition\, we show that every graph with $n$ vertices and average degree at least $n^{1-\frac{2}{s}+\varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{\varepsilon\,s}(1)$ vertices\, which is also optimal up to the constant hidden in the $O(.)$ notation\, and resolves a conjecture of Verstraëte. \nJoint work with Benny Sudakov and Istvan Tomon.
URL:https://dimag.ibs.re.kr/event/2023-05-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230509T163000
DTEND;TZID=Asia/Seoul:20230509T173000
DTSTAMP:20260506T071212
CREATED:20230413T233653Z
LAST-MODIFIED:20240707T073713Z
UID:7041-1683649800-1683653400@dimag.ibs.re.kr
SUMMARY:Jozef Skokan\, Separating the edges of a graph by a linear number of paths
DESCRIPTION:Recently\, Letzter proved that any graph of order n contains a collection P of $O(n \log^*n)$ paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f. We improve this upper bound to 19n\, thus answering a question of Katona and confirming a conjecture independently posed by Balogh\, Csaba\, Martin\, and Pluhar and by Falgas-Ravry\, Kittipassorn\, Korandi\, Letzter\, and Narayanan. \nOur proof is elementary and self-contained.
URL:https://dimag.ibs.re.kr/event/2023-05-09/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230502T163000
DTEND;TZID=Asia/Seoul:20230502T173000
DTSTAMP:20260506T071212
CREATED:20230315T140009Z
LAST-MODIFIED:20240705T164210Z
UID:6916-1683045000-1683048600@dimag.ibs.re.kr
SUMMARY:Rob Morris\, An exponential improvement for diagonal Ramsey
DESCRIPTION:The Ramsey number $R(k)$ is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of $K_k$. It has been known since the work of Erdős and Szekeres in 1935\, and Erdős in 1947\, that $2^{k/2} < R(k) < 4^k$\, but in the decades since the only improvements have been by lower order terms. In this talk I will sketch the proof of a very recent result\, which improves the upper bound of Erdős and Szekeres by a (small) exponential factor. \nBased on joint work with Marcelo Campos\, Simon Griffiths and Julian Sahasrabudhe.
URL:https://dimag.ibs.re.kr/event/2023-05-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230425T163000
DTEND;TZID=Asia/Seoul:20230425T173000
DTSTAMP:20260506T071212
CREATED:20230315T025543Z
LAST-MODIFIED:20240707T073739Z
UID:6905-1682440200-1682443800@dimag.ibs.re.kr
SUMMARY:Hyunwoo Lee (이현우)\, On perfect subdivision tilings
DESCRIPTION:For a given graph $H$\, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $\delta_{sub}(n\, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$\, we asymptotically determined the value of $\delta_{sub}(n\, H)$. More precisely\, for every graph $H$ with at least one edge\, there is a constant $1 < \xi^*(H)\leq 2$ such that $\delta_{sub}(n\, H) = \left(1 - \frac{1}{\xi^*(H)} + o(1) \right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise\, the threshold depends on the parity of $n$.
URL:https://dimag.ibs.re.kr/event/2023-04-25/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230411T163000
DTEND;TZID=Asia/Seoul:20230411T173000
DTSTAMP:20260506T071212
CREATED:20230220T150254Z
LAST-MODIFIED:20240705T165050Z
UID:6812-1681230600-1681234200@dimag.ibs.re.kr
SUMMARY:James Davies\, Two structural results for pivot-minors
DESCRIPTION:Pivot-minors can be thought of as a dense analogue of graph minors. We shall discuss pivot-minors and two recent results for proper pivot-minor-closed classes of graphs. In particular\, that for every graph H\, the class of graphs containing no H-pivot-minor is 𝜒-bounded\, and also satisfies the (strong) Erdős-Hajnal property.
URL:https://dimag.ibs.re.kr/event/2023-04-11/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230404T163000
DTEND;TZID=Asia/Seoul:20230404T173000
DTSTAMP:20260506T071212
CREATED:20230116T010354Z
LAST-MODIFIED:20240707T073814Z
UID:6667-1680625800-1680629400@dimag.ibs.re.kr
SUMMARY:István Tomon\, Configurations of boxes
DESCRIPTION:Configurations of axis-parallel boxes in $\mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their perceived simplicity\, there are many problems involving their structure that are not well understood. I will talk about a construction that shows that their structure might be more complicated than people conjectured.
URL:https://dimag.ibs.re.kr/event/2023-04-04/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230328T163000
DTEND;TZID=Asia/Seoul:20230328T173000
DTSTAMP:20260506T071212
CREATED:20230223T141945Z
LAST-MODIFIED:20240705T165050Z
UID:6831-1680021000-1680024600@dimag.ibs.re.kr
SUMMARY:Tianchi Yang\, On the maximum number of edges in k-critical graphs
DESCRIPTION:In this talk\, we will discuss the problem of determining the maximum number of edges in an n-vertex k-critical graph. A graph is considered k-critical if its chromatic number is k\, but any proper subgraph has a chromatic number less than k. The problem remains open for any integer k ≥ 4. Our presentation will showcase an improvement on previous results achieved by employing a combination of extremal graph theory and structural analysis. We will introduce a key lemma\, which may be of independent interest\, as it sheds light on the partial structure of dense k-critical graphs. This is joint work with Cong Luo and Jie Ma.
URL:https://dimag.ibs.re.kr/event/2023-03-28/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230321T163000
DTEND;TZID=Asia/Seoul:20230321T173000
DTSTAMP:20260506T071212
CREATED:20230204T071556Z
LAST-MODIFIED:20240705T170008Z
UID:6762-1679416200-1679419800@dimag.ibs.re.kr
SUMMARY:Younjin Kim (김연진)\, Problems on Extremal Combinatorics
DESCRIPTION:Extremal Combinatorics studies the maximum or minimum size of finite objects (numbers\, sets\, graphs) satisfying certain properties. In this talk\, I introduce the conjectures I solved on Extremal Combinatorics\, and also introduce recent extremal problems.
URL:https://dimag.ibs.re.kr/event/2023-03-21/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230314T163000
DTEND;TZID=Asia/Seoul:20230314T173000
DTSTAMP:20260506T071212
CREATED:20230120T011459Z
LAST-MODIFIED:20240707T073956Z
UID:6701-1678811400-1678815000@dimag.ibs.re.kr
SUMMARY:Stijn Cambie\, Recent progress on the Union-closed conjecture and related
DESCRIPTION:We give a summary on the work of the last months related to Frankl’s Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $\mathcal F$)\, then $\mathcal F$ contains an (abundant) element that belongs to at least half of the sets. Meanwhile\, there are many related versions of the conjecture due to Frankl. For example\, graph theorists may prefer the equivalent statement that every bipartite graph has a vertex that belongs to no more than half of the maximal independent sets. While even proving the ε-Union-Closed Sets Conjecture was out of reach\, Poonen and Cui & Hu conjectured already stronger forms. \nAt the end of last year\, progress was made on many of these conjectures. Gilmer proved the ε-Union-Closed Sets Conjecture using an elegant entropy-based method which was sharpened by many others. Despite a sharp approximate form of the union-closed conjecture as stated by Chase and Lovett\, a further improvement was possible. In a different direction\, Kabela\, Polak and Teska constructed union-closed family sets with large sets and few abundant elements. \nThis talk will keep the audience up-to-date and give them insight in the main ideas. \nPeople who would like more details\, can join the Ecopro-reading session on the 14th of March (10 o’clock\, B332) as well. Here we go deeper in the core of the proofs and discuss possible directions for the full resolution.
URL:https://dimag.ibs.re.kr/event/2023-03-14/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230307T163000
DTEND;TZID=Asia/Seoul:20230307T173000
DTSTAMP:20260506T071212
CREATED:20230111T063520Z
LAST-MODIFIED:20240707T074009Z
UID:6655-1678206600-1678210200@dimag.ibs.re.kr
SUMMARY:Eunjin Oh (오은진)\, Parameterized algorithms for the planar disjoint paths problem
DESCRIPTION:Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i\,t_i)_{i=1}^k$ of vertices\, the goal of the planar disjoint paths problem is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1\,\ldots\,k\}$. This problem has been studied extensively due to its numerous applications such as VLSI layout and circuit routing. However\, this problem is NP-complete even for grid graphs. This motivates the study of this problem from the viewpoint of parameterized algorithms. \nIn this talk\, I will present a $2^{O(k^2)}n$-time algorithm for the planar disjoint paths problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020]. \nThis is joint work with Kyungjin Cho and Seunghyeok Oh.
URL:https://dimag.ibs.re.kr/event/2023-03-07/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230228T163000
DTEND;TZID=Asia/Seoul:20230228T173000
DTSTAMP:20260506T071212
CREATED:20230213T001557Z
LAST-MODIFIED:20240707T074017Z
UID:6782-1677601800-1677605400@dimag.ibs.re.kr
SUMMARY:Maya Sankar\, The Turán Numbers of Homeomorphs
DESCRIPTION:Let $X$ be a 2-dimensional simplicial complex. Denote by $\text{ex}_{\hom}(n\,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of $\text{ex}_{\hom}(n\,X)$ have recently been determined for all surfaces $X$. I will discuss these results\, as well as ongoing work bounding $\text{ex}_{\hom}(n\,X)$ for arbitrary 2-dimensional simplicial complexes $X$.
URL:https://dimag.ibs.re.kr/event/2023-02-28/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230221T163000
DTEND;TZID=Asia/Seoul:20230221T173000
DTSTAMP:20260506T071212
CREATED:20230110T061742Z
LAST-MODIFIED:20240705T170041Z
UID:6636-1676997000-1677000600@dimag.ibs.re.kr
SUMMARY:Meike Hatzel\, Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parametrised by the Size of the Cutset: Twin-Width Meets Flow-Augmentation
DESCRIPTION:We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem\, given a directed graph $G$\, pairs of vertices (called terminals) $(s_1\,t_1)$\, $(s_2\,t_2)$\, and $(s_3\,t_3)$\, and an integer $k$\, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths\, all $s_2t_2$-paths\, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis\, Cygan\, Hajiaghayi\, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012\, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. \nOn the technical side\, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim\, Kratsch\, Pilipczuk\, Wahlström\, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width\, a recently introduced structural parameter [Bonnet\, Kim\, Thomassé\, Watrigant\, FOCS 2020]: By a recent characterization [Bonnet\, Giocanti\, Ossona de Mendez\, Simon\, Thomassé\, Toruńczyk\, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof\, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor\, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.
URL:https://dimag.ibs.re.kr/event/2023-02-21/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230214T163000
DTEND;TZID=Asia/Seoul:20230214T173000
DTSTAMP:20260506T071212
CREATED:20221122T113208Z
LAST-MODIFIED:20240705T171025Z
UID:6504-1676392200-1676395800@dimag.ibs.re.kr
SUMMARY:Raphael Steiner\, Strengthening Hadwiger's conjecture for 4- and 5-chromatic graphs
DESCRIPTION:Hadwiger’s famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29\, (1997)\, pp. 139-144] conjectured the following strengthening of Hadwiger’s conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G\, then G contains a $K_t$-minor rooted at S. We prove this conjecture in the first open case of t=4. Notably\, our result also directly implies a stronger version of Hadwiger’s conjecture for 5-chromatic graphs as follows: Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact\, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph. Joint work with Anders Martinsson (ETH).
URL:https://dimag.ibs.re.kr/event/2023-02-14/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230131T163000
DTEND;TZID=Asia/Seoul:20230131T173000
DTSTAMP:20260506T071212
CREATED:20230110T062223Z
LAST-MODIFIED:20240707T074122Z
UID:6639-1675182600-1675186200@dimag.ibs.re.kr
SUMMARY:Abhishek Methuku\, A proof of the Erdős–Faber–Lovász conjecture
DESCRIPTION:The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk\, I will sketch a proof of this conjecture for every large n. Joint work with D. Kang\, T. Kelly\, D. Kühn and D. Osthus.
URL:https://dimag.ibs.re.kr/event/2023-01-31/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230117T163000
DTEND;TZID=Asia/Seoul:20230117T173000
DTSTAMP:20260506T071212
CREATED:20221213T152329Z
LAST-MODIFIED:20240707T074134Z
UID:6556-1673973000-1673976600@dimag.ibs.re.kr
SUMMARY:Noleen Köhler\, Twin-Width VIII: Delineation and Win-Wins
DESCRIPTION:We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated by twin-width (or simply\, delineated) if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$\, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures of subclasses $\mathcal D$ of $\mathcal C$\, FO model checking on $\mathcal D$ is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT\, STOC ’22] and permutation graphs [BKTW\, JACM ’22] are effectively delineated\, while subcubic graphs are not. On the one hand\, we prove that interval graphs\, and even\, rooted directed path graphs are delineated. On the other hand\, we observe or show that segment graphs\, directed path graphs (with arbitrarily many roots)\, and visibility graphs of simple polygons are not delineated. \nIn an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not)\, we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW\, SODA ’21]. We show that $K_{t\,t}$-free segment graphs\, and axis-parallel $H_t$-free unit segment graphs have bounded twin-width\, where $H_t$ is the half-graph or ladder of height $t$. In contrast\, axis-parallel $H_4$-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated. \nMore broadly\, we explore which structures (large bicliques\, half-graphs\, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results\, combined with the FPT algorithm for first-order model checking on graphs given with $O(1)$-sequences [BKTW\, JACM ’22]\, give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If $p$ is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C\, then $p(G) \ge k$ can be decided in FPT time $f(k)\cdot |V (G)|O(1)$. For instance\, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains\, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width. \nJoint work with Édouard Bonnet\, Dibyayan Chakraborty\, Eun Jung Kim\, Raul Lopes and Stéphan Thomassé.
URL:https://dimag.ibs.re.kr/event/2023-01-17/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230110T163000
DTEND;TZID=Asia/Seoul:20230110T173000
DTSTAMP:20260506T071212
CREATED:20221123T222545Z
LAST-MODIFIED:20240705T171022Z
UID:6518-1673368200-1673371800@dimag.ibs.re.kr
SUMMARY:Mamadou Moustapha Kanté\, MSOL-Definable decompositions
DESCRIPTION:I will first introduce the notion of recognisability of languages of terms and then its extensions to sets of relational structures. In a second step\, I will discuss relations with decompositions of graphs/matroids and why their MSOL-definability is related to understanding recognisable sets. I will finally explain  how to define in MSOL branch-decompositions for finitely representable matroids of bounded path-width. This is joint work with Rutger Campbell\, Bruno Guillon\, Eun Jung Kim\, and Sang-il Oum.
URL:https://dimag.ibs.re.kr/event/2023-01-10/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230103T163000
DTEND;TZID=Asia/Seoul:20230103T173000
DTSTAMP:20260506T071212
CREATED:20221010T054006Z
LAST-MODIFIED:20240707T074146Z
UID:6280-1672763400-1672767000@dimag.ibs.re.kr
SUMMARY:Youngho Yoo (유영호)\, Approximating TSP walks in subcubic graphs
DESCRIPTION:The Graphic Travelling Salesman Problem is the problem of finding a spanning closed walk (a TSP walk) of minimum length in a given connected graph. The special case of the Graphic TSP on subcubic graphs has been studied extensively due to their worst-case behaviour in the famous $\frac{4}{3}$-integrality-gap conjecture on the “subtour elimination” linear programming relaxation of the Metric TSP. \nWe prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$\, confirming a conjecture of Dvořák\, Král’\, and Mohar. This bound is best possible and we characterize the extremal subcubic examples meeting this bound. We also give a quadratic time combinatorial algorithm to find such a TSP walk. In particular\, we obtain a $\frac{5}{4}$-approximation algorithm for the Graphic TSP on cubic graphs. Joint work with Michael Wigal and Xingxing Yu.
URL:https://dimag.ibs.re.kr/event/2023-01-03/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20221228T163000
DTEND;TZID=Asia/Seoul:20221228T173000
DTSTAMP:20260506T071212
CREATED:20221221T082326Z
LAST-MODIFIED:20240705T170042Z
UID:6590-1672245000-1672248600@dimag.ibs.re.kr
SUMMARY:Stijn Cambie\, The 69-conjecture and more surprises on the number of independent sets
DESCRIPTION:Various types of independent sets have been studied for decades. As an example\, the minimum number of maximal independent sets in a connected graph of given order is easy to determine (hint; the answer is written in the stars). When considering this question for twin-free graphs\, it becomes less trivial and one discovers some surprising behaviour. The minimum number of maximal independent sets turns out to be; \n\nlogarithmic in the number of vertices for arbitrary graphs\,\nlinear for bipartite graphs\nand exponential for trees.\n\nFinally\, we also have a sneak peek on the 69-conjecture\, part of an unpublished work on an inverse problem on the number of independent sets. \nIn this talk\, we will focus on the basic concepts\, the intuition behind the statements and sketch some proof ideas. \nThe talk is based on joint work with Stephan Wagner\, with the main chunk being available at arXiv:2211.04357.
URL:https://dimag.ibs.re.kr/event/2022-12-28/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20221206T163000
DTEND;TZID=Asia/Seoul:20221206T173000
DTSTAMP:20260506T071212
CREATED:20220908T152618Z
LAST-MODIFIED:20240707T074218Z
UID:6153-1670344200-1670347800@dimag.ibs.re.kr
SUMMARY:Giannos Stamoulis\, Model-Checking for First-Order Logic with Disjoint Paths Predicates in Proper Minor-Closed Graph Classes
DESCRIPTION:The disjoint paths logic\, FOL+DP\,  is an extension of First Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1\,y_1\,\ldots\,x_k\,y_k)\,$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i\,$ for $i\in \{1\,\ldots\, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every minor-closed graph class\, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP\, namely the scattered disjoint paths logic\, FOL+SDP\, where we further consider the atomic predicate $\mathsf{s-sdp}_k(x_1\,y_1\,\ldots\,x_k\,y_k)\,$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus.\nJoint work with Petr A. Golovach and Dimitrios M. Thilikos.
URL:https://dimag.ibs.re.kr/event/2022-12-06/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20221122T163000
DTEND;TZID=Asia/Seoul:20221122T173000
DTSTAMP:20260506T071212
CREATED:20221018T043943Z
LAST-MODIFIED:20240707T074455Z
UID:6362-1669134600-1669138200@dimag.ibs.re.kr
SUMMARY:Seonghyuk Im (임성혁)\, A proof of the Elliott-Rödl conjecture on hypertrees in Steiner triple systems
DESCRIPTION:A linear $3$-graph is called a (3-)hypertree if there exists exactly one path between each pair of two distinct vertices.  A linear $3$-graph is called a Steiner triple system if each pair of two distinct vertices belong to a unique edge. \nA simple greedy algorithm shows that every $n$-vertex Steiner triple system $G$ contains all hypertrees $T$ of order at most $\frac{n+3}{2}$. On the other hand\, it is not immediately clear whether one can always find larger hypertrees in $G$. In 2011\, Goodall and de Mier proved that a Steiner triple system $G$ contains at least one spanning tree. However\, one cannot expect the Steiner triple system to contain all possible spanning trees\, as there are many Steiner triple systems that avoid numerous spanning trees as subgraphs. Hence it is natural to wonder how much one can improve the bound from the greedy algorithm. \nIndeed\, Elliott and Rödl conjectured that an $n$-vertex Steiner triple system $G$ contains all hypertrees of order at most $(1-o(1))n$. We prove the conjecture by Elliott and Rödl. \nThis is joint work with Jaehoon Kim\, Joonkyung Lee\, and Abhishek Methuku.
URL:https://dimag.ibs.re.kr/event/2022-11-22/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
END:VCALENDAR