Discrete Mathematics Group

Hong Liu (刘鸿), A solution to Erdős and Hajnal's odd cycle problem
2020-12-08 16:30 - 17:30 (Asia/Seoul) | Room B232, IBS (기초과학연구원)
I will go over the history on the study of the set of cycle lengths of graphs with large average degree or chromatic number, and discuss recent work with Richard Montgomery on this topic. In particular, we will see the divergence of harmonic sum of odd cycle lengths in graphs with large chromatic number and the appearance of cycle lengths in very sparse sequences (such as powers of 2). The methods developed in this work allows also us to embed equally divided clique subdivisions, which was conjectured by Thomassen.

Deniz Sarikaya, What means Hamiltonicity for infinite graphs and how to force it via forbidden induced subgraphs
2020-12-03 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The study of Hamiltonian graphs, i.e. finite graphs having a cycle that contains all vertices of the graph, is a central theme of finite graph theory. For infinite graphs such a definition cannot work, since cycles are finite. We shall debate possible concepts of Hamiltonicity for infinite graphs and eventually follow the topological approach by Diestel and Kühn [2,3], which allows to generalize several results about being a Hamiltonian graph to locally finite graphs, i.e. graphs where each vertex has finite degree. An infinite cycle of a locally finite connected graph G is defined as a homeomorphic image of the unit circle $S^1$ in the Freudenthal compactification |G| of G. Now we call G Hamiltonian if there is an infinite cycle in |G| containing all vertices of G. For an introduction see [1].
We examine how to force Hamiltonicity via forbidden induced subgraphs and present recent extensions of results for Hamiltonicity in finite claw-free graphs to locally finite ones. The first two results are about claw- and net-free graphs, claw- and bull-free graphs, the last also about further graph classes being structurally richer, where we focus on paws as relevant subgraphs, but relax the condition of forbidding them as induced subgraphs.
The goal of the talk is twofold: (1) We introduce the history of the topological viewpoint and argue that there are some merits to it (2) sketch the proofs for the results mentioned above in some details.
This is based on joint work [4,5] with Karl Heuer.
Bibliography
[1] R. Diestel (2017) Infinite Graphs. In: Graph Theory. Graduate Texts in Mathematics, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53622-3_8
[2] R. Diestel and D. Kühn, On infinite cycles I, Combinatorica 24 (2004), pp. 69-89.
[3] R. Diestel and D. Kühn, On infinite cycles II, Combinatorica 24 (2004), pp. 91-116.
[4] K. Heuer and D. Sarikaya, Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: nets and bulls, arXiv:2006.09160
[5] K. Heuer and D. Sarikaya, Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs II: paws, arXiv:2006.09166

Joonkyung Lee (이준경), On common graphs
2020-12-02 17:00 - 18:00 | Zoom ID:8628398170 (123450)
A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdős, conjectured that every graph is common. The conjectures by Erdős and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s.
Despite its importance, the full classification of common graphs is still a wide open problem and has not seen much progress since the early 1990s. In this lecture, I will present some old and new techniques to prove whether a graph is common or not.

Debsoumya Chakraborti, Rainbow matchings in edge-colored simple graphs
2020-12-01 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
There has been much research on finding a large rainbow matching in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barát, Gyárfás, and Sárközy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not loops) with $2q$ colors where each color appears at least $q$ times, there is always a rainbow matching of size $q$. We prove that $2q + o(q)$ colors are enough if the graph is simple, confirming the conjecture asymptotically for simple graphs. We also make progress in the lower bound on the required number of colors for simple graphs, which disproves a conjecture of Aharoni and Berger. We use a randomized algorithm to obtain a large rainbow matching, and the analysis of the algorithm is based on differential equations method. We will also briefly comment on the limitations of using our probabilistic approach for the problem. This talk will be based on a joint work with Po-Shen Loh.

Joonkyung Lee (이준경), On Ramsey multiplicity
2020-11-30 17:00 - 18:00 | Zoom ID:8628398170 (123450)
Ramsey’s theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey’s theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the Ramsey multiplicity $M(H;n)$ be the minimum number of labelled copies of monochromatic $H$ over all 2-edge-colouring of $K_n$. We define the Ramsey multiplicity constant $C(H)$ is defined by $C(H):=lim_{nrightarrowinfty}frac{M(H,n)}{n(n-1)cdots(n-v+1)}$. I will discuss various bounds for C(H) that are known so far.

Da Qi Chen, Bipartite Saturation
2020-11-26 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
In extremal graph theory, a graph G is H-saturated if G does not contain a copy of H but adding any missing edge to G creates a copy of H. The saturation number, sat(n, H), is the minimum number of edges in an n-vertex H-saturated graph. This class of problems was first studied by Zykov and Erdős, Hajnal, and Moon. They also determined the saturation number when H is a clique and classified the extremal structures.
In this talk, we will focus mainly on the bipartite saturation problem (which was also first introduced by Erdős, Hajnal, and Moon). Here, we always assume that both G and H are bipartite graphs. Then, G is H-saturated if G does not contain H but adding any missing edge across the bipartition creates a copy of H. We can then similarly define sat(n, H) as the minimum number of edges of an n-by-n bipartite graph that is also H-saturated. One of the most interesting and natural questions here is to determine the saturation number for the complete bipartite graph $K_{s, t}$. When s=t, the saturation number and its extremal structures were determined long ago but nothing else is known for the general case. Half a decade ago, Gan, Korandi, and Sudakov gave an asymptotically tight bound that was only off by an additive constant. We will highlight the main ideas behind that proof and show, with some additional techniques, how the bound can be improved to achieve tightness for the case when s=t-1.
This talk is based on collaborative work with Debsoumya Chakraborti and Mihir Hasabnis. See arXiv: https://arxiv.org/abs/2009.07651 for the full paper.

Duksang Lee (이덕상), Characterizing matroids whose bases form graphic delta-matroids
2020-11-24 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements. This is joint work with Sang-il Oum.

Yijia Chen (陈翌佳), Graphs of bounded shrub-depth, through a logic lens
2020-11-19 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
Shrub-depth is a graph invariant often considered as an extension
of tree-depth to dense graphs. In this talk I will explain our recent
proofs of two results about graphs of bounded shrub-depth.

Every graph property definable in monadic-second order logic,
e.g., 3-colorability, can be evaluated by Boolean circuits of constant
depth and polynomial size, whose depth only depends on the
shrub-depth of input graphs.
Graphs of bounded shrub-depth can be characterized by
a finite set of forbidden induced subgraphs [Ganian et al. 2015].

Central to the first result is the definability in first-order logic of
tree-models for graphs of bounded shrub-depth. For the second
result, we observe that shrub-depth can be easily generalized
to infinite graphs, and thus some classical tools, i.e., Craig’s
Interpolation and Łoś-Tarski Theorem, in model theory are
applicable to graphs of bounded shrub-depth.
This is joint work with Jörg Flum.

Meike Hatzel, Constant congestion bramble
2020-11-11 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
In this talk I will present a small result we achieved during a workshop in February this year. My coauthors on this are Marcin Pilipczuk, Paweł Komosa and Manuel Sorge.
A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) cap V(H_2)
eq emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble.
Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $Omega(n^{1/2+delta})$ requires size exponential in $Omega(n^{2delta})$ for any fixed $delta in (0,frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx, and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $widetilde{Omega}(k^{1/2})$ and size $widetilde{O}(k^{3/2})$. ($widetilde{Omega}$ and $widetilde{O}$ hide polylogarithmic factors and divisors, respectively.)
We first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $widetilde{Omega}(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $delta in (0,frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $widetilde{Omega}(k^{1/2+delta})$ and size $2^{widetilde{O}(k^{2delta})}$.

Casey Tompkins, Extremal forbidden poset problems in Boolean and linear lattices
2020-11-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Extending the classical theorem of Sperner on the maximum size of an antichain in the Boolean lattice, Katona and Tarján introduced a general extremal function $La(n,P)$, defined to be the maximum size of a family of subsets of $[n]$ which does not contain a given poset $P$ among its containment relations. In this talk, I will discuss what is known about the behavior of $La(n,P)$ and its natural extension to the lattice of subspaces of a vector space over a finite field. In particular, I will highlight some recent joint work with Jimeng Xiao. Many open problems will also be discussed.

Daniel Cranston, Vertex Partitions into an Independent Set and a Forest with Each Component Small
2020-11-05 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
For each integer $kge 2$, we determine a sharp bound on
$operatorname{mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$, where $I$ is an independent set and $G $ is a forest in which each component has at most k vertices. For each $k$ we construct an infinite family of examples showing our result is best possible. Hendrey, Norin, and Wood asked for the largest function $g(a,b)$ such that if $operatorname{mad}(G) < g(a,b)$ then $V(G)$ has a partition into sets $A$ and $B$ such that $operatorname{mad}(G[A]) < a$ and $operatorname{mad}(G[B]) < b$. They specifically asked for the value of $g(1,b)$, which corresponds to the case that $A$ is an independent set. Previously, the only values known were $g(1,4/3)$ and $g(1,2)$. We find the value of $g(1,b)$ whenever $4/3 < b < 2$. This is joint work with Matthew Yancey.

Jaeseong Oh (오재성), A 2-isomorphism theorem for delta-matroids
2020-11-03 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Whitney’s 2-Isomorphism Theorem characterises when two graphs have isomorphic cycle matroids. In this talk, we present an analogue of this theorem for graphs embedded in surfaces by characterising when two graphs in surface have isomorphic delta-matroids. This is based on the joint work with Iain Moffatt.

Jeong Ok Choi (최정옥), Various game-theoretic models on graphs
2020-10-27 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We introduce some of well-known game-theoretic graph models and related problems.
A contagion game model explains how an innovation diffuses over a given network structure and focuses on finding conditions on which structure an innovation becomes epidemic. Regular infinite graphs are interesting examples to explore. We show that regular infinite trees make an innovation least advantageous to be epidemic considering the whole class of infinite regular graphs.
A network creation game model, on the other hand, tries to explain the dynamics on forming a network structure when each vertex plays independently and selfishly. An important question is how costly a formation can be made without any central coordination, and the concept of Price of Anarchy (PoA) is introduced. In the model originally suggested by Fabrikant et al., PoA measures how bad the forming cost can be at Nash equilibria compared to absolute minimum, and they conjectured that this inefficiency can happen only when some tree structures are formed (Tree Conjecture). We will introduce recent progress on this tree conjecture, remaining open problems, and possible variations.
This talk includes part of joint work with Unjong Yu.

Chun-Hung Liu (劉俊宏), Asymptotic dimension of minor-closed families and beyond
2020-10-22 10:10 - 11:10 | Zoom ID:95464969835 (356260)
The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance functions in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.

Joonkyung Lee (이준경), On graph norms for complex-valued functions
2020-10-21 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For any given graph $H$, one may define a natural corresponding functional $|.|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $alpha$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $|.|_H$ (resp. there is $alpha$ such that $|.|_{H,alpha}$) is a norm on the vector space of real-valued (resp. complex-valued) functions. This generalises Gowers norms, a widely used tool in extremal combinatorics to quantify quasirandomness.
We unify these two seemingly different notions of graph norms in real- and complex-valued settings, by proving that $H$ is complex-norming if and only if it is real-norming. Our proof does not explicitly construct a suitable $2$-edge-colouring $alpha$ but obtain its existence and uniqueness, which may be of independent interest.
As an application, we give various example graphs that are not norming. In particular, we show that hypercubes are not norming, which answers the only question appeared in Hatami’s pioneering work in the area that remained untouched. This is joint work with Alexander Sidorenko.

Hong Liu (刘鸿), Cycles and trees in graphs (8/8)
2020-10-15 17:00 - 18:00 | Zoom ID:8628398170 (123450)
This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.

Hong Liu (刘鸿), Cycles and trees in graphs (7/8)
2020-10-14 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (6/8)
2020-10-13 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (5/8)
2020-10-12 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (4/8)
2020-10-08 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (3/8)
2020-10-07 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (2/8)
2020-10-06 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Hong Liu (刘鸿), Cycles and trees in graphs (1/8)
2020-10-05 17:00 - 18:00 | Zoom ID:8628398170 (123450)

Minki Kim (김민기), Complexes of graphs with bounded independence number
2020-09-29 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $G$ be a graph on $V$ and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose faces are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of $I_n(G)$ for various classes of graphs, focusing on the class of graphs with bounded maximum degree. This is joint work with Alan Lew.

Zihan Tan, Towards Tight(er) Bounds for the Excluded Grid Theorem
2020-09-24 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f$, such that for every integer $g > 0$, every graph of treewidth at least $f(g)$ contains the g×g-grid as a minor. For every integer $g > 0$, let $f(g)$ be the smallest value for which the theorem holds. Establishing tight bounds on $f(g)$ is an important graph-theoretic question. Robertson and Seymour showed that f(g) is at least of order $g^2 log g$. For a long time, the best known upper bounds on $f(g)$ were super-exponential in $g$. The first polynomial upper bound of $f(g) = O(g^{98} operatorname{poly log} g)$ was proved by Chekuri and Chuzhoy. It was later improved to $f(g) = O(g^{36} operatorname{poly log} g)$, and then to $f(g) = O(g^{19} operatorname{poly log} g)$. In this talk, we present our recent work that further improves this bound to $f(g) = O(g^9 operatorname{poly log} g)$ via a simpler proof. Moreover, while there are natural barriers that seem to prevent the previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this talk can lead to even tighter bounds on $f(g)$.
This talk is based on joint work with Julia Chuzhoy.

Jinha Kim (김진하), Collapsibility of Non-Cover Complexes of Graphs
2020-09-22 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $G$ be a graph on the vertex set $V$. A vertex subset $W subset V$ is a cover of $G$ if $V setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex on $V$ whose faces are non-covers of $G$. Then the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. In this talk, I will show the $(|V(G)|-igamma(G)-1)$-collapsibility of the non-cover complex of a graph $G$ where $igamma(G)$ denotes the independence domination number of $G$ using the minimal exclusion sequence method. This is joint work with Ilkyoo Choi and Boram Park.

Luke Postle, Further progress towards Hadwiger's conjecture
2020-09-17 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $tge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(tsqrt{log t})$ and hence is $O(tsqrt{log t})$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is $O(t(log t)^{beta})$-colorable for every $beta > 1/4$, making the first improvement on the order of magnitude of the $O(tsqrt{log t})$ bound. Here we show that every graph with no $K_t$ minor is $O(t (log t)^{beta})$-colorable for every $beta > 0$; more specifically, they are $O(t (log log t)^{6})$-colorable.

Debsoumya Chakraborti, Maximum number of cliques in a graph with bounded maximum degree
2020-09-15 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Generalized extremal problems have been one of the central topics of study in extremal combinatorics throughout the last few decades. One such simple-looking problem, maximizing the number of cliques of a fixed order in a graph with a fixed number of vertices and given maximum degree, was recently resolved by Chase. Settling a conjecture of Kirsch and Radcliffe, we resolve the edge variant of this problem, where the number of edges is fixed instead of the number of vertices. This talk will be based on joint work with Da Qi Chen.

Sebastian Siebertz, Rank-width meets stability
2020-09-10 17:10 - 18:10 | Zoom ID: 869 4632 6610 (ibsdimag)
Forbidden graph characterizations provide a convenient way of specifying graph classes, which often exhibit a rich combinatorial and algorithmic theory. A prime example in graph theory are classes of bounded tree-width, which are characterized as those classes that exclude some planar graph as a minor. Similarly, in model theory, classes of structures are characterized by configurations that are forbidden as logical interpretations or transductions. Two notions from classical model theory are (monadic) stability and (monadic) dependence, which correspond to the impossibility of interpreting with first-order logic (after a vertex coloring step) arbitrary long linear orders and all graphs, respectively. Examples of monadically stable classes of graphs are nowhere dense graph classes, and examples of monadically dependent classes are classes of bounded rank-width (or equivalently, bounded clique-width), which can be seen as a dense analog of classes of bounded tree-width.
I will give an overview over recent approaches to combine model theoretic and graph theoretic tools to derive structural and algorithmic results for classes of (finite) graphs. I assume no background from logic.

Rutger Campbell, Disasters in abstracting combinatorial properties of linear dependence
2020-09-08 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let E be a finite set and I be a collection of subsets of E. When is there a set of real vectors indexed by E such that I correspond to its linearly independent subsets? In 1935, Whitney introduced matroids using some necessary conditions for this. However, complete characterizations with various techniques are intractable. This remains the case even if it is already known that there is a set of complex vectors indexed by E whose collection of linearly independent subsets corresponds to I.
