Seminars and Colloquia by Series

A proof of the Erdős–Faber–Lovász conjecture and related problems

Series
Graph Theory Seminar
Time
Tuesday, December 14, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Abhishek MethukuUniversity of Birmingham

Please Note: Note the unusual time!

The famous 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 briefly sketch a proof of this conjecture for every large n. If time permits, I will also talk about our solution to a problem of Erdős from 1977 about chromatic index of hypergraphs with bounded codegree. Joint work with D. Kang, T. Kelly, D.Kuhn and D. Osthus.

Density and graph edge coloring

Series
Graph Theory Seminar
Time
Tuesday, December 7, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guangming JingAugusta University

Given a multigraph $G=(V,E)$, the chromatic index $\chi'(G)$ is the minimum number of colors needed to color the edges of $G$ such that no two incident edges receive the same color. Let $\Delta(G)$ be the maximum degree of $G$ and let  $\Gamma(G):=\max \big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm
{\rm and \hskip 2mm odd} \big\}$. $\Gamma(G)$ is called the density of $G$. Clearly, the density is a lower bound for the chromatic index $\chi'(G)$. Moreover, this value can be computed in polynomial time. Goldberg and Seymour in the 1970s conjectured that $\chi'(G)=\lceil\Gamma(G)\rceil$ for any multigraph $G$ with $\chi'(G)\geq\Delta(G)+2$, known as the Goldberg-Seymour conjecture. In this talk we will discuss this conjecture and some related open problems. This is joint work with Guantao Chen and Wenan Zang.

Constructions in combinatorics via neural networks

Series
Graph Theory Seminar
Time
Tuesday, November 30, 2021 - 12:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Adam Zsolt WagnerTel Aviv University

Please Note: Note the unusual time!

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 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.

Strong 4-colourings of graphs

Series
Graph Theory Seminar
Time
Tuesday, November 23, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Jessica McDonaldAuburn University

In this talk we’ll discuss strong 4-colourings of graphs and prove two new cases of the Strong Colouring Conjecture. Let H be a graph with maximum degree at most 2, and let G be obtained from H by gluing in vertex-disjoint copies of K_4. We’ll show that if H contains at most one odd cycle of length exceeding 3, or if H contains at most 3 triangles, then G is 4-colourable. This is joint work with Greg Puleo.

Irregular $\mathbf{d_n}$-Process is distinguishable from Uniform Random $\mathbf{d_n}$-graph

Series
Graph Theory Seminar
Time
Tuesday, November 16, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Erlang SuryaGeorgia Institute of Technology

For a graphic degree sequence $\mathbf{d_n}= (d_1 , . . . , d_n)$ of graphs with vertices $v_1 , . . . , v_n$, $\mathbf{d_n}$-process is the random graph process that inserts one edge at a time at random with the restriction that the degree of $v_i$ is at most $d_i$ . In 1999, N. Wormald asked whether the final graph of random $\mathbf{d_n}$-process is "similar" to the uniform random graph with degree sequence $\mathbf{d_n}$ when $\mathbf{d_n}=(d,\dots, d)$. We answer this question for the $\mathbf{d_n}$-process when the degree sequence $\mathbf{d_n}$ that is not close to being regular. We used the method of switching for stochastic processes; this allows us to track the edge statistics of the $\mathbf{d_n}$-process. Joint work with Mike Molloy and Lutz Warnke.

Counting paths, cycles, and other subgraphs in planar graphs

Series
Graph Theory Seminar
Time
Tuesday, November 9, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ryan MartinIowa State University

For a planar graph $H$, let ${\bf N}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The case where $H$ is the path on $3$ vertices, $H=P_3$, was established by Alon and Caro. The case of $H=P_4$ was determined, also exactly, by Gy\H{o}ri, Paulos, Salia, Tompkins, and Zamora. In this talk, we will give the asymptotic values for $H$ equal to $P_5$ and $P_7$ as well as the cycles $C_6$, $C_8$, $C_{10}$ and $C_{12}$ and discuss the general approach which can be used to compute the asymptotic value for many other graphs $H$. This is joint work with Debarun Ghosh, Ervin Győri, Addisu Paulos, Nika Salia, Chuanqi Xiao, and Oscar Zamora and also joint work with Chris Cox.

Line transversals in families of connected sets in the plane

Series
Graph Theory Seminar
Time
Tuesday, November 2, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Shira ZerbibIowa State University

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that under the same condition there are four lines intersecting all the sets. We also prove a colorful version of this result under weakened conditions on the sets, improving results of Holmsen from 2013. Our proofs use the topological KKM theorem. Joint with Daniel McGinnis.

Geometric bijections between subgraphs and orientations of a graph

Series
Graph Theory Seminar
Time
Tuesday, October 26, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Zoom
Speaker
Changxin DingBrandeis University

Please Note: Zoom link: https://us04web.zoom.us/j/77238664391 Password: graphs!

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called geometric bijections. Recently we have extended the geometric bijections to  subgraph-orientation correspondences. In this talk, I will introduce the bijections and the geometry behind them.

 

Counting colorings of triangle-free graphs

Series
Graph Theory Seminar
Time
Tuesday, October 19, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruijia CaoGeorgia Institute of Technology

Please Note: Note the unusual time!

In this talk, we will discuss the main results of our paper, Counting Colorings of Triangle-Free Graphs, in which we prove the Johansson-Molloy theorem for the upper bound on the chromatic number of a triangle free graph using a novel counting approach developed by Matthieu Rosenfeld, and also extend this result to obtain a lower bound on the number of proper q-colorings for a triangle free graph.  The talk will go over the history of the problem, an outline of our approach, and a high-level sketch of the main proofs. This is joint work with Anton Bernshteyn, Tyler Brazelton, and Akum Kang.

Turán numbers of some complete degenerate hypergraphs

Series
Graph Theory Seminar
Time
Tuesday, October 5, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaofan YuanGeorgia Institute of Technology

Please Note: Note the unusual time!

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It is well-known in the graph case that $ex(n,K_{s,t})=\Theta(n^{2-1/s})$ when $t$ is sufficiently larger than $s$. We generalize the above to hypergraphs by showing that if $s_r$ is sufficiently larger than $s_1,s_2,\cdots,s_{r-1}$ then $$ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})=\Theta\left(n^{r-\frac{1}{s_1s_2\cdots s_{r-1}}}\right).$$ This is joint work with Jie Ma and Mingwei Zhang.

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