## Seminars and Colloquia by Series

### Perfect matchings in random hypergraphs

Series
Graph Theory Seminar
Time
Tuesday, October 13, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Matthew KwanStanford University

For positive integers $d < k$ and $n$ divisible by $k$, let $m_d(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_1(2,n) = \lceil n/2\rceil$. However, in general, our understanding of the values of $m_d(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In the first part of this talk, we discuss a new "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and "not too small" edge probability $p$ typically has the property that every spanning subgraph with minimum $d$-degree at least $(1+\varepsilon)m_d(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_d(k,n)$ without actually knowing its value.

The ideas in our work are quite powerful and can be applied to other problems: in the second part of this talk we highlight a recent application of these ideas to random designs, proving that a random Steiner triple system typically admits a decomposition of almost all its triples into perfect matchings (that is to say, it is almost resolvable).

Joint work with Asaf Ferber.

### Inducibility of graphs and tournaments

Series
Graph Theory Seminar
Time
Tuesday, October 6, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Florian PfenderUniversity of Colorado Denver

A classical question in extremal graph theory asks to maximize the number of induced copies of a given graph or tournament in a large host graph, often expressed as a density. A simple averaging argument shows that the limit of this density exists as the host graph is allowed to grow. Razborov's flag algebra method is well suited to generate bounds on these quantities with the help of semidefinite programming. We will explore this method for a few small examples, and see how to modify it to fit our questions. The extremal graphs show some beautiful structures, sometimes fractal like, sometimes quasi random and sometimes even a combination of both.

### Breaking the degeneracy barrier for coloring graphs with no $K_t$ minors

Series
Graph Theory Seminar
Time
Tuesday, September 15, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Zi-Xia SongUniversity of Central Florida

Hadwiger's conjecture from 1943 states that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the early 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable.  In this talk, we show 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 Kostochka-Thomason bound.

This is joint work with  Sergey Norin and Luke Postle.

### Unavoidable dense induced subgraphs

Series
Graph Theory Seminar
Time
Tuesday, September 8, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/681348075/???? (replace ???? with password). For password, please email Anton Bernshteyn (bahtoh~at~gatech.edu)
Speaker
Rose McCartyUniversity of Waterloo

Thomassen conjectures that every graph of sufficiently large average degree has a subgraph of average degree at least d and girth at least k, for any d and k. What if we want the subgraph to be induced? Large cliques and bicliques are the obvious obstructions; we conjecture there are no others. We survey results in this direction, and we prove that every bipartite graph of sufficiently large average degree has either K_{d,d} or an induced subgraph of average degree at least d and girth at least 6.

### Saturation problems in Ramsey theory, ordered sets and geometry

Series
Graph Theory Seminar
Time
Tuesday, September 1, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/681348075/????. (replace ???? with password) For password, please email Anton Bernshteyn (bahtoh~at~gatech.edu)
Speaker
Zhiyu WangGeorgia Tech

A graph G is F-saturated if G is F-free and G+e is not F-free for any edge not in G. The saturation number of F, is the minimum number of edges in an n-vertex F-saturated graph. We consider analogues of this problem in other settings.  In particular we prove saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we only require that any extension of the combinatorial structure creates new copies of the forbidden configuration.  In this setting, we prove a semisaturation version of the Erdös-Szekeres theorem on convex k-gons, as well as multiple semisaturation theorems for sequences and posets. Joint work with Gábor Damásdi, Balázs Keszegh, David Malec, Casey Tompkins, and Oscar Zamora.

### Distributed algorithms and infinite graphs

Series
Graph Theory Seminar
Time
Tuesday, August 25, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/954562826
Speaker
Anton BernshteynGeorgia Tech

In the last twenty or so years, a rich theory has emerged concerning combinatorial problems on infinite graphs endowed with extra structure, such as a topology or a measure. It turns out that there is a close relationship between this theory and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. In this talk I will outline this relationship and present a number of applications.

### Anti-Ramsey number of edge-disjoint rainbow spanning trees

Series
Graph Theory Seminar
Time
Thursday, April 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhiyu WangUniversity of South Carolina

An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. The \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [{\em J. Graph Theory, 2016}] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. We show their conjecture is true and also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$. Joint work with Linyuan Lu.

### Counting critical subgraphs in k-critical graphs

Series
Graph Theory Seminar
Time
Thursday, October 3, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jie MaUniversity of Science and Technology of China

A graph is $k$-critical if its chromatic number is $k$ but any its proper subgraph has chromatic number less than $k$. Let $k\geq 4$. Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that every such graph contains $\Omega(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles. We mainly focus on 4-critical graphs and develop some novel tools for counting cycles of specified parity. Our main result shows that any $4$-critical graph on $n$ vertices contains $\Omega(n^2)$ odd cycles, which is tight up to a constant factor by infinite many graphs. As a crucial step, we prove the same bound for 3-connected non-bipartite graphs, which may be of independent interest. Using the tools, we also give a very short proof to the Gallai's problem for the case $k=4$. Moreover, we improve the longstanding lower bound of Abbott and Zhou to $\Omega(n^{1/(k-2)})$ for the general case $k\geq 5$. Joint work with Tianchi Yang.

### Quasirandom permutations

Series
Graph Theory Seminar
Time
Friday, September 13, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Dan KralMasaryk University and University of Warwick

A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. For example, it is well-known that a graph G with edge-density p is quasirandom if and only if the density of C_4 in G is p^4+o(p^4); this property is known to equivalent to several other properties that hold for truly random graphs.  A similar phenomenon was established for permutations: a permutation is quasirandom if and only if the density of every 4-point pattern (subpermutation) is 1/4!+o(1).  We strengthen this result by showing that a permutation is quasirandom if and only if the sum of the densities of eight specific 4-point patterns is 1/3+o(1). More generally, we classify all sets of 4-point patterns having such property.

The talk is based on joint work with Timothy F. N. Chan, Jonathan A. Noel, Yanitsa Pehova, Maryam Sharifzadeh and Jan Volec.

### Independent set permutations, and matching permutations

Series
Graph Theory Seminar
Time
Thursday, April 18, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
David GalvinUniversity of Notre Dam
To any finite real sequence, we can associate a permutation $\pi$, via: $\pi(k)$ is the index of the $k$th smallest element of the sequence. This association was introduced in a 1987 paper of Alavi, Malde, Schwenk and Erd\H{o}s, where they used it to study the possible patterns of rises and falls that can occur in the matching sequence of a graph (the sequence whose $k$th term is the number of matchings of size $k$), and in the independent set sequence. The main result of their paper was that {\em every} permutation can arise as the independent set permutation'' of some graph. They left open the following extremal question: for each $n$, what is the smallest order $m$ such that every permutation of $[n]$ can be realized as the independent set permutation of some graph of order at most $m$? We answer this question. We also improve Alavi et al.'s upper bound on the number of permutations that can be realized as the matching permutation of some graph. There are still many open questions in this area. This is joint work with T. Ball, K. Hyry and K. Weingartner, all at Notre Dame.