Discrete Mathematics Group

Joonkyung Lee (이준경), Majority dynamics on sparse random graphs
2021-10-12 16:30 - 17:30 (Asia/Seoul) | Room B232, IBS (기초과학연구원)
Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph $G(n,p)$, the random initial $pm 1$-assignment converges to a $99%$-agreement with high probability whenever $p=omega(1/n)$.
This conjecture was first confirmed for $pgeqlambda n^{-1/2}$ for a large constant $lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< lambda n^{-1/2}$. We break this $Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $lambda’ n^{-3/5}log n leq p leq lambda n^{-1/2}$ with a large constant $lambda’>0$.
Joint work with Debsoumya Chakraborti, Jeong Han Kim and Tuan Tran.

Paul Seymour, Polynomial bounds for chromatic number
2021-10-08 10:00 - 11:00 | Zoom ID: 869 4632 6610 (ibsdimag)
The Gyárfás-Sumner conjecture says that for every forest $H$, there is a function $f$ such that the chromatic number $chi(G)$ is at most $f(omega(G))$ for every $H$-free graph $G$ (“$H$-free” means with no induced subgraph isomorphic to $H$, and $omega(G)$ is the size of the largest clique of $G$). This well-known conjecture has been proved only for a few types of forest.
Nevertheless, there is a much stronger conjecture, due to Esperet: that for every forest $H$, there is a polynomial function $f$ as above. As one might expect, this has been proved for even fewer types of forest; and the smallest tree $H$ for which Esperet’s conjecture is not known is the five-vertex path $P_5$.
A third notorious conjecture is the Erdős-Hajnal conjecture, that for every graph $H$, there exists $c>0$ such that $alpha(G)omega(G)ge |G|^c$ for every $H$-free graph $G$ (where $alpha(G)$ is the size of the largest stable set of $G$). The smallest graph $H$ for which this is not known is also $P_5$, which, conveniently, is a forest; and every forest that satisfies Esperet’s conjecture also satisfies the Erdős-Hajnal conjecture. So there is substantial interest in the chromatic numbers of $P_5$-free graphs. The best upper bound that was previously known, due to Esperet, Lemoine, Maffray, and Morel, was that $chi(G)le (5/27)3^omega(G)$ for every $P_5$-free graph $G$ with $omega(G) > 2$. In recent work with Alex Scott and Sophie Spirkl, we have proved several results related to Esperet’s conjecture, including proofs of its truth for some new types of forest $H$, and a “near-polynomial” bound when $H = P_5$, that $chi(G) le omega(G)^{log_2(omega(G))}$ for every $P_5$-free graph $G$ with $omega(G) > 2$. We survey these results and give a proof of the new bound for $P_5$.

Eunjin Oh (오은진), Feedback Vertex Set on Geometric Intersection Graphs
2021-10-05 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(sqrt{k}log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].

Péter Pál Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities
2021-09-30 16:30 - 17:30 | Zoom ID: 869 4632 6610 (ibsdimag)
The Alon-Jaeger-Tarsi conjecture states that for any finite field $mathbb{F}$ of size at least 4 and any nonsingular matrix $M$ over $mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Mx$ has a 0 component. In joint work with János Nagy we proved this conjecture when the size of the field is sufficiently large, namely, when $61

Kevin Hendrey, Extremal functions for sparse minors
2021-09-28 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, [c(H) = (1+o(1))max (v(H)/2, v(H)-alpha(H)),] extending recent results of Haslegrave, Kim and Liu (2020). Joint work with Sergey Norin and David R. Wood.

Dabeen Lee (이다빈), Mixing sets, submodularity, and chance-constrained optimization
2021-09-07 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this talk, we first explain basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull. This is based on joint work with Fatma Fatma Kılınç-Karzan and Simge Küçükyavuz.

Cheolwon Heo (허철원), Representations of even-cycle matroids
2021-08-31 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A signed graph is a pair $(G,Sigma)$ where $G$ is a graph and $Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,Sigma)$ if $|C cap Sigma|$ is even; otherwise, $C$ is odd. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,Sigma)$. For even-cycle matroids, two fundamental questions arise:
(1) what is the relationship between two signed graphs representing the same even-cycle matroids?
(2) how many signed graphs can an even-cycle matroid have?
For (a), we characterize two signed graphs $(G_1,Sigma_1)$ and $(G_2,Sigma_2)$ where $G_1$ and $G_2$ are $4$-connected that represent the same even-cycle matroids.
For (b), we introduce pinch-graphic matroids, which can generate exponentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a pair of vertices such that every odd cycle intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. This is joint work with Bertrand Guenin and Irene Pivotto.

Eun Jung Kim (김은정), A Constant-factor Approximation for Weighted Bond Cover
2021-08-24 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The Weighted $mathcal F$-Vertex Deletion for a class $mathcal F$ of graphs asks, given a weighted graph $G$, for a minimum weight vertex set $S$ such that $G-Sinmathcal F$. The case when $mathcal F$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $mathcal F$-Vertex Deletion. Only three cases of minor-closed $mathcal F$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set.
We study the problem for the class $mathcal F$ of $theta_c$-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA’14] which states the following: any graph $G$ containing a $theta_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $theta_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $mathcal F$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
This is joint work with Euiwoong Lee and Dimitrios M. Thilikos.

Petr Hliněný, Twin-width is linear in the poset width
2021-08-18 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced quite recently, in 2020 by Bonnet, Kim, Thomassé, and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing a prior result on FO model checking of posets of bounded width in FPT. However, their translation from poset width to twin-width was indirect and giving only a very loose double-exponential bound.
We prove that posets of width d have twin-width at most 9d with a direct and elementary argument, and show that this bound is tight up to a constant factor. Specifically, for posets of width 2, we prove that in the worst case their twin-width is also equal to 2. These two theoretical results are complemented with straightforward algorithms to construct the respective contraction sequence for a given poset.
(Joint work with my student Jakub Balaban who obtained the main ideas in his bachelor thesis.)

Linda Cook, Two results on graphs with holes of restricted lengths
2021-08-17 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc provided a structural description of the class of even-hole-free graphs. I will describe the structure of all graphs that contain only holes of length $ell$ for every $ell geq 7$ (joint work with Jake Horsfield, Myriam Preissmann, Paul Seymour, Ni Luh Dewi Sintiari, Cléophée Robin, Nicolas Trotignon, and Kristina Vuškovíc.
Analysis of how holes interact with graph structure has yielded detection algorithms for holes of various lengths and parities. In 1991, Bienstock showed it is NP-Hard to test whether a graph G has an even (or odd) hole containing a specified vertex $v in V(G)$. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc gave a polynomial-time algorithm to recognize even-hole-free graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi, and Seymour provided a simpler and slightly faster algorithm to test whether a graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour, and Spirkl provided a polynomial-time algorithm to test whether a graph contains an odd hole. Later that year, Chudnovsky, Scott, and Seymour strengthened this result by providing a polynomial-time algorithm to test whether a graph contains an odd hole of length at least $ell$ for any fixed integer $ell geq 5$. I will present a polynomial-time algorithm (joint work with Paul Seymour) to test whether a graph contains an even hole of length at least $ell$ for any fixed integer $ell geq 4$.

Duksang Lee (이덕상), Intertwining connectivities for vertex-minors and pivot-minors
2021-08-10 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.

Maria Chudnovsky, Induced subgraphs and tree decompositions
2021-07-28 15:00 - 16:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Tree decompositions are a powerful tool in structural graph theory; they are traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has until recently remained out of reach.
Tree decompositions are closely related to the existence of “laminar collections of separations” in a graph, which roughly means that the separations in the collection “cooperate” with each other, and the pieces that are obtained when the graph is simultaneously decomposed by all the separations in the collection “line up” to form a tree structure. Such collections of separations come up naturally in the context of forbidden minors.
In the case of families where induced subgraphs are excluded, while there are often natural separations, they are usually very far from forming a laminar collection. In what follows we mostly focus on families of graphs of bounded degree. It turns out that due to the bound on the degree, these collections of natural separations can be partitioned into a bounded number of laminar collections. This in turn allows to us obtain a wide variety of structural and algorithmic results, which we will survey in this talk.

Euiwoong Lee (이의웅), The Karger-Stein algorithm is optimal for k-cut
2021-07-27 15:00 - 16:00 | Room B232, IBS (기초과학연구원)
In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. It is easy to see that the elegant randomized contraction algorithm of Karger and Stein for global mincut (k=2) can be naturally extended for general k with the running time of $O(n^{2k-2})$. Using tree packings, Thorup gave a deterministic algorithm that has the same running time.
In this work, we show that for any fixed $kge 2$, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least $Omega(n^{-k})$. This also gives an extremal bound of $O(n^k)$ on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are essentially tight. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most $(2-delta)OPT/k$, using the Sunflower lemma.
Joint work with Anupam Gupta, David G. Harris, and Jason Li.

Semin Yoo (유세민), Combinatorics of Euclidean spaces over finite fields
2021-07-20 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients, denoted by $binom{n}{k}_{q}$, which give the number of $k$-dimensional subspaces in $mathbb{F}_{q}^{n}$. When $q$ goes to $1$, this reverts to the binomial coefficients which measure the number of $k$-sets in $left [ n right ]$.
In this talk, we add one more structure in $mathbb{F}_{q}^{n}$, which is the Euclidean quadratic form: $text{dot}_{n}:=x_{1}^{2}+x_{2}^{2}+cdots+x_{n}^{2}$. It turns out that the number of quadratic subspaces of Euclidean type in $(mathbb{F}_{q}^{n},text{dot}_{n})$ can be described as the form of the analogue of binomial coefficients. The main goal of this talk is to define the dot-analogues of the binomial coefficients and to study related combinatorics. No prior knowledge about the theory of quadratic form is required.

Stefan Weltge, Integer programs with bounded subdeterminants and two nonzeros per row
2021-07-14 17:00 - 18:00 | Room B232, IBS (기초과학연구원)
We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k=0 (bipartite graphs) and for k=1.
We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to b-matching.
This is joint work with Samuel Fiorini, Gwenaël Joret, and Yelena Yuditsky.

Jaehoon Kim (김재훈), $K_{r+1}$-saturated graphs with small spectral radius
2021-07-13 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $ein E(overline G)$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length 2 in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $nto infty$.

Suil O (오수일), Eigenvalues and [a, b]-factors in regular graphs
2021-07-06 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
For positive integers, $r ge 3, h ge 1,$ and $k ge 1$, Bollobás, Saito, and Wormald proved some sufficient conditions for an $h$-edge-connected $r$-regular graph to have a k-factor in 1985. Lu gave an upper bound for the third-largest eigenvalue in a connected $r$-regular graph to have a $k$-factor in 2010. Gu found an upper bound for certain eigenvalues in an $h$-edge-connected $r$-regular graph to have a $k$-factor in 2014. For positive integers $a le b$, an even (or odd) $[a, b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v in V (G)$, $d_H(v)$ is even (or odd) and $a le d_H(v) le b$. In this talk, we provide best upper bounds (in terms of $a, b$, and $r$) for certain eigenvalues (in terms of $a, b, r$, and $h$) in an $h$-edge-connected $r$-regular graph $G$ to guarantee the existence of an even $[a, b]$-factor or an odd $[a, b]$-factor. This result extends the one of Bollobás, Saito, and Wormald, the one of Lu, and the one of Gu.

Florian Gut and Attila Joó, Large vertex-flames in uncountable digraphs
2021-06-30 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
The local connectivity $ kappa_D(r,v) $ from $ r $ to $ v $ is defined to be the maximal number of internally disjoint $rrightarrow v $ paths in $ D $. A spanning subdigraph $ L $ of $ D $ with $ kappa_L(r,v)=kappa_D(r,v) $ for every $ vin V-r $ must have at least $ sum_{vin V-r}kappa_D(r,v) $ edges. It was shown by Lovász that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $ L $ can be captured by the following characterization: For every $ vin V-r $ there is a system $ mathcal{P}_v $ of internally disjoint $ rrightarrow v $ paths in $ L $ covering all the ingoing edges of $ v $ in $ L $ such that one can choose from each $ Pin mathcal{P}_v $ either an edge or an internal vertex in such a way that the resulting set meets every $ rrightarrow v $ path of $ D $. We prove that every digraph of size at most $ aleph_1 $ admits such a spanning subdigraph $ L $. The question if this remains true for larger digraphs remains open.

Jeong Ok Choi (최정옥), Invertibility of circulant matrices of arbitrary size
2021-06-29 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
In this talk, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations in terms of the entries in the first row with integer coefficients. Using these conditions we show the invertibility of some family of circulant matrices with particular forms of integers generated by a primitive element in $mathbb{Z}_p$. Also, the invertibility of circulant 0, 1-matrices can be argued combinatorially by applying sufficient conditions. This is joint work with Youngmi Hur.

Hongseok Yang (양홍석), DAG-symmetries and Symmetry-Preserving Neural Networks
2021-06-22 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
The preservation of symmetry is one of the key tools for designing data-efficient neural networks. A representative example is convolutional neural networks (CNNs); they preserve translation symmetries, and this symmetry preservation is often attributed to their success in real-world applications. In the machine-learning community, there is a growing body of work that explores a new type of symmetries, both discrete and continuous, and studies neural networks that preserve those symmetries.
In this talk, I will explain what I call DAG-symmetries and our preliminary results on the shape of neural networks that preserve these symmetries. DAG-symmetries are finite variants of DAG-exchangeability developed by Jung, Lee, Staton, and Yang (2020) in the context of probabilistic symmetries. Using these symmetries, we can express that when a neural network works on, for instance, sets of bipartite graphs whose edges are labelled with reals, the network depends on neither the order of elements in the set nor the identities of vertices of the graphs. I will explain how a group of specific DAG-symmetries is constructed by applying a form of wreath product over a given finite DAG. Then, I will explain what linear layers of neural networks preserving these symmetries should look like.
This is joint work with Dongwoo Oh.

Alan Lew, Representability and boxicity of simplicial complexes
2021-06-16 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
An interval graph is the intersection graph of a family of intervals in the real line. Motivated by problems in ecology, Roberts defined the boxicity of a graph G to be the minimal k such that G can be written as the intersection of k interval graphs.
A natural higher-dimensional generalization of interval graphs is the class d-representable complexes. These are simplicial complexes that carry the information on the intersection patterns of a family of convex sets in $mathbb R^d$. We define the d-boxicity of a simplicial complex X to be the minimal k such that X can be written as the intersection of k d-representable complexes.
A classical result of Roberts, later rediscovered by Witsenhausen, asserts that the boxicity of a graph with n vertices is at most n/2. Our main result is the following high dimensional extension of Roberts’ theorem: Let X be a simplicial complex on n vertices with minimal non-faces of dimension at most d. Then, the d-boxicity of X is at most $frac{1}{d+1}binom{n}{d}$.
Examples based on Steiner systems show that our result is sharp. The proofs combine geometric and topological ideas.

O-joung Kwon (권오정), Classes of intersection digraphs with good algorithmic properties
2021-06-08 15:00 - 16:00 | Zoom ID: 875 9395 3555 (relay) [CLOSED]
An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_vcap T_v
eq emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed.
Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width, we introduce its directed analogue called `bi-mim-width’ and prove that various classes of reflexive intersection digraphs have bounded bi-mim-width. In particular, we show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear mim-width of $H$-graphs [On the Tractability of Optimization Problems on $H$-Graphs. Algorithmica 2020]
For applications, we introduce a novel framework of directed versions of locally checkable problems, that streamlines the definitions and the study of many problems in the literature and facilitates their common algorithmic treatment. We obtain unified polynomial-time algorithms for these problems on digraphs of bounded bi-mim-width, when a branch decomposition is given. Locally checkable problems include Kernel, Dominating Set, and Directed $H$-Homomorphism.
This seminar is a part of the
“Round the World Relay in Combinatorics”
on June 8, 2021.
http://people.maths.ox.ac.uk/scott/relay.htm

2:00 UTC, 11:00 KST Melbourne (Australia) Monash University
David Wood (Monash University), Universality in Minor-Closed Graph Classes
3:00 UTC, 12:00 KST Shanghai (China) Shanghai Center for Mathematical Sciences
Baogang Xu (Nanjing Normal University, China), On coloring of graphs of girth 2l+1 without longer odd holes
4:00 UTC, 13:00 KST Auckland (New Zealand) The University of Auckland
Jeroen Schillewaert (University of Auckland), Constructing highly regular expanders from hyperbolic Coxeter groups
5:00 UTC, 14:00 KST Sydney (Australia) Combinatorial Mathematics Society of Australasia
Gordon Royle (University of Western Australia (UWA)), Real chromatic roots of planar graphs
6:00 UTC, 15:00 KST Daejeon (Korea) IBS Discrete Mathematics Group
O-joung Kwon (Incheon National University and IBS Discrete Mathematics Group), Classes of intersection digraphs with good algorithmic properties
7:00 UTC, 16:00 KST Krakow (Poland) Jagiellonian University
Bartosz Walczak (Jagiellonian), Coloring polygon visibility graphs and their generalizations
8:00 UTC, 17:00 KST Glasgow (UK) University of Glasgow
Heng Guo (University of Edinburgh), A Markov chain approach towards the sampling Lovász local lemma
9:00 UTC, 18:00 KST London (UK) London School of Economics
Annika Heckel (LMU), How does the chromatic number of a random graph vary?
10:00 UTC, 19:00 KST Moscow (Russia) Moscow Institute of Physics and Technology
Noga Alon (Princeton and Tel Aviv)
11:00 UTC, 20:00 KST Budapest (Hungary) Hungarian Academy of Sciences + Eötvös Loránd University
László Lovász (Eotvos University, Budapest)
12:00 UTC, 21:00 KST Bordeaux (France) LaBRI
Carla Groenland (Utrecht University), Universal Graphs and Labelling Schemes
13:00 UTC, 22:00 KST New York (USA) City University of New York + Montclair State University + Hofstra University
Deepak Bal (Montclair State University)
14:00 UTC, 23:00 KST Prague (Czech) Czech Academy of Sciences + Czech Technical University + London School of Economics
Dhruv Mubayi (University of Illinois at Chicago), The feasible region of hypergraphs
15:00 UTC, 00:00 KST Brno (Czech) Masaryk University
James Davies (University of Waterloo), Colouring circle graphs and their generalisations
16:00 UTC, 01:00 KST Oxford (UK) University of Oxford
Jacob Fox (Stanford University), Removal lemmas
17:00 UTC, 02:00 KST Columbus (USA) Ohio State University
Allan Sly (Princeton University)
18:00 UTC, 03:00 KST Rio (Brazil) Instituto de Matemática Pura e Aplicada
Marcelo Campos (IMPA), The singularity probability of a random symmetric matrix is exponentially small
19:00 UTC, 04:00 KST Atlanta (USA) Georgia Institute of Technology
Jim Geelen (University of Waterloo), Homomorphisms and colouring for graphs and binary matroids
20:00 UTC, 05:00 KST Santiago (Chile) Universidad de Chile
David Conlon (Caltech)
21:00 UTC, 06:00 KST Burnaby (Canada) Simon Fraser University
Fan Chung (UCSD), Trees and forests in Green’s functions of a graph
22:00 UTC, 07:00 KST Victoria (Canada) University of Victoria
Andrew Suk (UCSD), Turán-type problems for point-line incidences
23:00 UTC, 08:00 KST Fairbanks (USA) University of Alaska
Ron Gould (Emory), Chorded cycles

Adam Zsolt Wagner, Constructions in combinatorics via neural networks
2021-06-02 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk, I will give a very basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the “game” of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.

Doowon Koh (고두원), Mattila-Sjölin type functions: A finite field model
2021-06-01 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
Let $mathbb{F}_q$ be a finite field of order $q$ which is a prime power. In the finite field setting, we say that a function $phicolon mathbb{F}_q^dtimes mathbb{F}_q^dto mathbb{F}_q$ is a Mattila-Sjölin type function in $mathbb{F}_q^d$ if for any $Esubset mathbb{F}_q^d$ with $|E|gg q^{frac{d}{2}}$, we have $phi(E, E)=mathbb{F}_q$. The main purpose of this talk is to present the existence of such a function. More precisely, we will construct a concrete function $phi: mathbb{F}_q^4times mathbb{F}_q^4to mathbb{F}_q$ with $qequiv 3 mod{4}$ such that if $Esubset mathbb F_q^4$ with $|E|>q^2,$ then $phi(E,E)=mathbb F_q$. This is a joint work with Daewoong Cheong, Thang Pham, and Chun-Yen Shen.

Dimitrios M. Thilikos, Bounding Obstructions sets: the cases of apices of minor closed classes
2021-05-26 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Given a minor-closed graph class ${cal G}$, the (minor) obstruction of ${cal G}$ is the set of all minor-minimal graphs not in ${cal G}$. Given a non-negative integer $k$, we define the $k$-apex of ${cal A}$ as the class containing every graph $G$ with a set $S$ of vertices whose removal from $G$ gives a graph on ${cal G}$. We prove that every obstruction of the $k$-apex of ${cal G}$ has size bounded by some 4-fold exponential function of $p(k)$ where p is a polynomial function whose degree depends on the size of the minor-obstructions of ${cal G}$. This bound drops to a 2-fold exponential one when ${cal G}$ excludes some apex graph as a minor (i.e., a graph in the $1$-apex of planar graphs).
Joint work with Ignasi Sau and Giannos Stamoulis.

Ben Lund, Limit shape of lattice Zonotopes
2021-05-25 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A convex lattice polytope is the convex hull of a set of integral points. Vershik conjectured the existence of a limit shape for random convex lattice polygons, and three proofs of this conjecture were given in the 1990s by Bárány, by Vershik, and by Sinai. To state this old result more precisely, there is a convex curve $L subset [0,1]^2$ such that the following holds. Let $P$ be a convex lattice polygon chosen uniformly at random from the set of convex lattice polygons with vertices in $[0,N]^2$. Then, for $N$ sufficiently large, $(1/N)P$ will be arbitrarily close (in Hausdorff distance) to $L$ with high probability. It is an open question whether there exists a limit shape for three dimensional polyhedra.
I will discuss this problem and some relatives, as well as joint work with Bárány and Bureaux on the existence of a limit shape for lattice zonotopes in all dimensions.

Benjamin Bumpus, Directed branch-width: A directed analogue of tree-width
2021-05-21 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Many problems that are NP-hard in general become tractable on `structurally recursive’ graph classes. For example, consider classes of bounded tree- or clique-width. Since the 1990s, many directed analogues of tree-width have been proposed. However, many natural problems (e.g. directed HamiltonPath and MaxCut) remain intractable on such digraph classes of `bounded width’.
In this talk, I will introduce a new tree-width analogue for digraphs called directed branch-width which allows us to define digraph classes for which many problems (including directed HamiltonPath and MaxCut) become linear-time solvable. Furthermore, via the definition of directed branch-width, I will obtain a generalisation to digraphs of Gurski and Wanke’s characterization of graph classes of bounded tree-width in terms of their line graphs.
This is joint work with Kitty Meeks and William Pettersson.

Pascal Gollin, Enlarging vertex-flames in countable digraphs
2021-05-18 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by Joó and the analogue of Lovász’ result for countable digraphs was shown.
In this talk, I present a strengthening of this result stating that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.
Joint work with Joshua Erde and Attila Joó.

Johannes Carmesin, A Whitney type theorem for surfaces: characterising graphs with locally planar embeddings
2021-05-12 17:00 - 18:00 | Zoom ID: 869 4632 6610 (ibsdimag)
Given a graph, how do we construct a surface so that the graph embeds in that surface in an optimal way? Thomassen showed that for minimum genus as optimality criterion, this problem would be NP-hard. Instead of minimum genus, here we use local planarity — and provide a polynomial algorithm.
Our embedding method is based on Whitney’s trick to use matroids to construct embeddings in the plane. Consequently we obtain a characterisation of the graphs admitting locally planar embeddings in surfaces in terms of a certain matroid being co-graphic.

Mark Siggers, The list switch homomorphism problem for signed graphs
2021-05-11 16:30 - 17:30 | Room B232, IBS (기초과학연구원)
A signed graph is a graph in which each edge has a positive or negative sign. Calling two graphs switch equivalent if one can get from one to the other by the iteration of the local action of switching all signs on edges incident to a given vertex, we say that there is a switch homomorphism from a signed graph $G$ to a signed graph $H$ if there is a sign preserving homomorphism from $G’$ to $H$ for some graph $G’$ that is switch equivalent to $G$. By reductions to CSP this problem, and its list version, are known to be either polynomial time solvable or NP-complete, depending on $H$. Recently those signed graphs $H$ for which the switch homomorphism problem is in $P$ were characterised. Such a characterisation is yet unknown for the list version of the problem.
We talk about recent work towards such a characterisation and about how these problems fit in with bigger questions that still remain around the recent CSP dichotomy theorem.
