Topological bounds for graph representations over any field

Frédéric Meunier

École Nationale des Ponts et Chaussées, Paris

2019/11/21 Thu 4:30PM-5:30PM, Room B232, IBS (기초과학연구원)

ABSTRACT

Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We [...]

The post Frédéric Meunier, Topological bounds for graph representations over any field appeared first on KAIST Discrete Math Seminar.

]]>Topological bounds for graph representations over any field

Frédéric Meunier

École Nationale des Ponts et Chaussées, Paris

École Nationale des Ponts et Chaussées, Paris

2019/11/21 Thu 4:30PM-5:30PM, Room B232, IBS (기초과학연구원)

ABSTRACT

Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.

This is joint work with Meysam Alishahi.

The post Frédéric Meunier, Topological bounds for graph representations over any field appeared first on KAIST Discrete Math Seminar.

]]>A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

Pascal Gollin

IBS Discrete Mathematics Group

2019/10/29 Tue 4:30PM-5:30PM Room B232, IBS (기초과학연구원)

Given a cardinal $\lambda$, a $\lambda$-packing of a graph $G$ is a family of $\lambda$ many edge-disjoint spanning trees of $G$, and a $\lambda$-covering of $G$ is a family of spanning trees covering $E(G)$.

We [...]

The post Pascal Gollin, A Cantor-Bernstein-type theorem for spanning trees in infinite graphs appeared first on KAIST Discrete Math Seminar.

]]>A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

Pascal Gollin

IBS Discrete Mathematics Group

IBS Discrete Mathematics Group

2019/10/29 Tue 4:30PM-5:30PM Room B232, IBS (기초과학연구원)

Given a cardinal $\lambda$, a $\lambda$-packing of a graph $G$ is a family of $\lambda$ many edge-disjoint spanning trees of $G$, and a $\lambda$-covering of $G$ is a family of spanning trees covering $E(G)$.

We show that if a graph admits a $\lambda$-packing and a $\lambda$-covering then the graph also admits a decomposition into $\lambda$ many spanning trees. In this talk, we concentrate on the case of $\lambda$ being an infinite cardinal. Moreover, we will provide a new and simple proof for a theorem of Laviolette characterising the existence of a $\lambda$-packing, as well as for a theorem of Erdős and Hajnal characterising the existence of a $\lambda$-covering.

Joint work with Joshua Erde, Attila Joó, Paul Knappe and Max Pitz.

The post Pascal Gollin, A Cantor-Bernstein-type theorem for spanning trees in infinite graphs appeared first on KAIST Discrete Math Seminar.

]]>First-order interpretations of bounded expansion classes

Jakub Gajarský

Technische Universität Berlin

2019/12/10 Tue 4:30PM-5:30PM (Room B232, IBS)

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim [...]

The post Jakub Gajarský, First-order interpretations of bounded expansion classes appeared first on KAIST Discrete Math Seminar.

]]>First-order interpretations of bounded expansion classes

Jakub Gajarský

Technische Universität Berlin

Technische Universität Berlin

2019/12/10 Tue 4:30PM-5:30PM (Room B232, IBS)

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

The post Jakub Gajarský, First-order interpretations of bounded expansion classes appeared first on KAIST Discrete Math Seminar.

]]>Induced Turán problems for hypergraphs

Ruth Luo

University of California, San Diego

2019/11/19 Tue 4:30PM-5:30PM (Room B232, IBS)

Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that [...]

The post Ruth Luo, Induced Turán problems for hypergraphs appeared first on KAIST Discrete Math Seminar.

]]>Induced Turán problems for hypergraphs

Ruth Luo

University of California, San Diego

University of California, San Diego

2019/11/19 Tue 4:30PM-5:30PM (Room B232, IBS)

Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that for all $xy \in E(F)$, $f(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs. Joint work with Zoltan Füredi.

The post Ruth Luo, Induced Turán problems for hypergraphs appeared first on KAIST Discrete Math Seminar.

]]>Stable sets in graphs with bounded odd cycle packing number

Tony Huynh

Monash University

2019/11/12 Tue 4:30PM-5:30PM (Room B232, IBS)

It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs. The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for [...]

The post Tony Huynh, Stable sets in graphs with bounded odd cycle packing number appeared first on KAIST Discrete Math Seminar.

]]>Stable sets in graphs with bounded odd cycle packing number

Tony Huynh

Monash University

Monash University

2019/11/12 Tue 4:30PM-5:30PM (Room B232, IBS)

It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs. The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for graphs without two disjoint odd cycles. The complexity of the stable set problem for graphs without $k$ disjoint odd cycles is a long-standing open problem for all other values of $k$. We prove that under the additional assumption that the input graph is embedded in a surface of bounded genus, there is a polynomial-time algorithm for each fixed $k$. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This result may be of independent interest and extends a theorem of Kawarabayashi and Nakamoto asserting that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface. Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which we prove to be efficiently solvable in our case. This is joint work with Michele Conforti, Samuel Fiorini, Gwenaël Joret, and Stefan Weltge.

The post Tony Huynh, Stable sets in graphs with bounded odd cycle packing number appeared first on KAIST Discrete Math Seminar.

]]>Sun Kim (김선)

Mathematical Institute, University of Cologne, Cologne, Germany

2019/11/05 Tue 4:30PM-5:30PM (room 1401, E6-1, KAIST)

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three [...]

The post Sun Kim (김선), Two identities in Ramanujan’s Lost Notebook with Bessel function series appeared first on KAIST Discrete Math Seminar.

]]>Sun Kim (김선)

Mathematical Institute, University of Cologne, Cologne, Germany

Mathematical Institute, University of Cologne, Cologne, Germany

2019/11/05 Tue 4:30PM-5:30PM (room 1401, E6-1, KAIST)

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three different interpretations for the double series, and showed that they are intimately connected with the classical circle and divisor problems in number theory. Furthermore, we established many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.

The post Sun Kim (김선), Two identities in Ramanujan’s Lost Notebook with Bessel function series appeared first on KAIST Discrete Math Seminar.

]]>On some properties of graph norms

Joonkyung Lee (이준경)

Department of Mathematics, University of Hamburg, Germany

2019/10/22 Tue 4:30PM-5:30PM (Room B232, IBS)

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t_H(W)$. One may then define corresponding functionals $\|W\|_{H}:=|t_H(W)|^{1/e(H)}$ [...]

The post Joonkyung Lee (이준경), On some properties of graph norms appeared first on KAIST Discrete Math Seminar.

]]>On some properties of graph norms

Joonkyung Lee (이준경)

Department of Mathematics, University of Hamburg, Germany

Department of Mathematics, University of Hamburg, Germany

2019/10/22 Tue 4:30PM-5:30PM (Room B232, IBS)

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t_H(W)$. One may then define corresponding functionals $\|W\|_{H}:=|t_H(W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t_H(|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ is a norm. We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of $\|.\|_{H}$. We also prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.

The post Joonkyung Lee (이준경), On some properties of graph norms appeared first on KAIST Discrete Math Seminar.

]]>Ramsey numbers of cycles under Gallai colorings

Zi-Xia Song (宋梓霞)

University of Central Florida

2019/10/15 Tue 4:30PM-5:30PM (Room B232, IBS)

For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy [...]

The post Zi-Xia Song (宋梓霞), Ramsey numbers of cycles under Gallai colorings appeared first on KAIST Discrete Math Seminar.

]]>Ramsey numbers of cycles under Gallai colorings

Zi-Xia Song (宋梓霞)

University of Central Florida

University of Central Florida

2019/10/15 Tue 4:30PM-5:30PM (Room B232, IBS)

For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4 $ vertices. For odd cycles, Bondy and Erd\H{o}s in 1973 conjectured that for all $k\ge1$ and $n\ge2$, $R_k(C_{2n+1})=n\cdot 2^k+1$. Recently, this conjecture has been verified to be true for all fixed $k$ and all $n$ sufficiently large by Jenssen and Skokan; and false for all fixed $n$ and all $k$ sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of $R_k(C_{2n})$ in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all $k$ and all $n$ under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings. Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.

The post Zi-Xia Song (宋梓霞), Ramsey numbers of cycles under Gallai colorings appeared first on KAIST Discrete Math Seminar.

]]>Reconstructing graphs from smaller subgraphs

Alexandr V. Kostochka

University of Illinois at Urbana-Champaign

2019/10/10 Thu 4:30PM-5:30PM

A graph or graph property is $\ell$-reconstructible if it is determined by the multiset of all subgraphs obtained by deleting $\ell$ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each $\ell\in\mathbb N$, there is an [...]

The post Alexandr V. Kostochka, Reconstructing graphs from smaller subgraphs appeared first on KAIST Discrete Math Seminar.

]]>Reconstructing graphs from smaller subgraphs

Alexandr V. Kostochka

University of Illinois at Urbana-Champaign

University of Illinois at Urbana-Champaign

2019/10/10 Thu 4:30PM-5:30PM

A graph or graph property is $\ell$-reconstructible if it is determined by the multiset of all subgraphs obtained by deleting $\ell$ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each $\ell\in\mathbb N$, there is an integer $n=n(\ell)$ such that every graph with at least $n$ vertices is $\ell$-reconstructible. We show that for each $n\ge7$ and every $n$-vertex graph $G$, the degree list and connectedness of $G$ are $3$-reconstructible, and the threshold $n\geq 7$ is sharp for both properties. We also show that all $3$-regular graphs are $2$-reconstructible.

The post Alexandr V. Kostochka, Reconstructing graphs from smaller subgraphs appeared first on KAIST Discrete Math Seminar.

]]>On Ramsey-type problems for paths and cycles in dense graphs

Alexandr V. Kostochka

University of Illinois at Urbana-Champaign

2019/10/08 Tue 4:30PM-5:30PM

A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More [...]

The post Alexandr V. Kostochka, On Ramsey-type problems for paths and cycles in dense graphs appeared first on KAIST Discrete Math Seminar.

]]>On Ramsey-type problems for paths and cycles in dense graphs

Alexandr V. Kostochka

University of Illinois at Urbana-Champaign

University of Illinois at Urbana-Champaign

2019/10/08 Tue 4:30PM-5:30PM

A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph $G$ arrows a graph $H$ if for any coloring of the edges of $G$ with two colors, there is a monochromatic copy of $H$. In these terms, the above puzzle claims that the complete $6$-vertex graph $K_6$ arrows the complete $3$-vertex graph $K_3$. We consider sufficient conditions on the dense host graphs $G$ to arrow long paths and even cycles. In particular, for large $n$ we describe all multipartite graphs that arrow paths and cycles with $2n$ edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such $n$. Also for large $n$ we find which minimum degree in a $(3n-1)$-vertex graph $G$ guarantees that $G$ arrows the $2n$-vertex path. This yields a more recent conjecture of Schelp. This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu.

The post Alexandr V. Kostochka, On Ramsey-type problems for paths and cycles in dense graphs appeared first on KAIST Discrete Math Seminar.

]]>