## Seminars and Colloquia by Series

### 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

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.

### Counting comparisons in the Erdős–Szekeres theorem

Series
Graph Theory Seminar
Time
Tuesday, September 28, 2021 - 15:45 for
Location
Skiles 005
Speaker
Misha LavrovKennesaw State University

This talk is motivated by the Erdős–Szekeres theorem on monotone subsequences: given a sequence of $rs+1$ distinct numbers, there is either a subsequence of $r+1$ of them in increasing order, or a subsequence of $s+1$ of them in decreasing order.

We'll consider many related questions with an algorithmic flavor, such as: if we want to find one of the subsequences promised, how many comparisons do we need to make? What if we have to pre-register our comparisons ahead of time? Does it help if we search a longer sequence instead?

Some of these questions are still open; some of them have answers. The results I will discuss are joint work with Jozsef Balogh, Felix Clemen, and Emily Heath at UIUC.

### The feasible region of induced graphs

Series
Graph Theory Seminar
Time
Tuesday, September 21, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xizhi LiuUniversity of Illinois at Chicago

Fix a graph $F$. A classical problem in extremal graph theory asks about how many induced copies of $F$ can a graph with edge density $\rho$ have? The only case in which we know the asymptotic solution is when $F$ is a complete graph, and it was solved completely only recently by Reiher using the flag algebra machinery. We will consider the other cases and show some results when $F$ is a complete bipartite graph or a complete graph minus one edge. Many interesting related open problems will also be introduced. Joint work with Dhruv Mubayi and Christian Reiher.

### Induced subgraphs and treewidth

Series
Graph Theory Seminar
Time
Tuesday, September 14, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Sophie SpirklUniversity of Waterloo

Treewidth, introduced by Robertson and Seymour in the graph minors series, is a fundamental measure of the complexity of a graph. While their results give an answer to the question, “what minors occur in graphs of large treewidth?,” the same question for induced subgraphs is still open. I will talk about some conjectures and recent results in this area. Joint work with Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Pawel Rzazewski, Kristina Vuskovic.

### Polynomial $\chi$-binding functions for $t$-broom-free graphs

Series
Graph Theory Seminar
Time
Tuesday, September 7, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joshua SchroederGeorgia Institute of Technology

For any positive integer $t$, a $t$-broom is a graph obtained from $K_{1,t+1}$ by subdividing an edge once.  In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$, where  $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of  Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to $7.5\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and {$t$-broom, $K_{t,t}$}-free graphs, we improve the bound to $o(\omega^{t-1+\frac{2}{t+1}})$. Joint work with Xiaonan Liu, Zhiyu Wang, and Xingxing Yu.

### Long cycles in essentially 4-connected projective-planar graphs

Series
Graph Theory Seminar
Time
Tuesday, August 31, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael WigalGeorgia Institute of Technology

Tutte paths have a critical role in the study of Hamiltonicity for 4-connected planar and other graph classes. We show quantitative Tutte path results in which we bound the number of bridges of the path. A corollary of this result is near optimal circumference bounds for essentially 4-connected planar and projective-planar graphs. Joint work with Xingxing Yu.