### 2021–2022 Academic Year

**Bill Kay**, Pacific Northwest National Laboratory

- Apr 29
- 2:30pm

**Abstract**: In this talk, I will discuss how discrete geometry can play a role in data classification.
Namely, I will introduce a data augmentation technique which provably reduces misclassification
error without increasing training time for linear classifiers, and discuss the efficacy
of this technique empirically on other kinds of classifiers. The talk will include
a gentle introduction to relevant data classification discrete geometry methods involved,
so no specific prior knowledge is required.

**David Galvin**, University of Notre Dame

- April 15
- 2:30pm

**Abstract**: Morse theory seeks to understand the shape of a manifold by studying smooth functions
on it. Though the objects involved are inherently continuous, the study of Morse functions
can lead to interesting and non-trivial combinatorial questions.

**Heather Blake, **Davidson College

- April 8
- 2:30pm

**Abstract**: Given a problem and a set of feasible solutions to that problem, the associated *reconfiguration problem* involves determining whether one feasible solution can be transformed to a different
feasible solution through a sequence of allowable moves, with the condition that the
intermediate stages are also feasible solutions.

Any reconfiguration problem can be modelled with a *reconfiguration graph*, where the vertices represent the feasible solutions and two vertices are adjacent
if and only if the corresponding feasible solutions can be transformed to each other
via *one* allowable move.

Our interest is in reconfiguring dominating sets of graphs. The *domination reconfiguration graph* of a graph \(G\), denoted \({\mathcal D}(G)\), has a vertex corresponding to each
dominating set of \(G\) and two vertices of \({\mathcal D}(G)\) are adjacent if and
only if the corresponding dominating sets differ by the deletion or addition of a
single vertex.

We are interested in properties of domination reconfiguration graphs. For example, it is easy to see that they are always connected and bipartite. While none has a Hamilton cycle, we explore families of graphs whose reconfiguration graphs have Hamilton paths.

**Gregory Clark**, University of Oxford

- April 1
- 2:30pm

**Abstract**: Measuring and interpreting the centrality of vertices in a graph is critical to numerous
problems. Recently, the principal eigenvector of a hypergraph has received increasing
attention from academics and practitioners alike. In particular, hypergraph network
models exhibit desirable properties such as admitting heterogeneous data types. Despite
the increasing attention, there is still hesitance for adopting these models. One
reason for this is a fundamental question: are the insights gained from a hypergraph
‘distinct enough’ from the graph case to warrant a new methodology? We present evidence
in the affirmative by considering the spectral ranking of a hypergraph and its shadow;
that is, the ranking of vertices according to the principal eigenvector of the associated
hypermatrix and the traditional co-occurrence matrix. To this end, we present the
first known example of a hypergraph whose most central node varies under the shadow
operation. We explore this topic by presenting several examples and results which
highlight the mechanisms underpinning this phenomenon.

**Hays Whitlatch, **Gonzaga University

- March 25
- 2:30pm

**Abstract**: Motivated by the question of computing the probability of successful power domination
by placing k monitors uniformly at random, in this presentation we give a recursive
formula to count the number of power domination sets of size k in a labeled complete
m-ary tree. As a corollary we show that the desired probability can be computed in
exponential with linear exponent time. This is joint work with Kat Shultis (Gonzaga
University) and three undergraduate students: Lana Kniahnitskaya, Michele Ortiz, and
Olivia Ramirez.

**Jozsef Balogh, **University of Illinois Urbana-Champaign

- March 15
- 2:30pm

**Abstract**: A classical combinatorial number theory problem is to determine the maximum size of
a Sidon set of \(\{ 1, 2, \ldots, n\}\), where a subset of integers is {\it Sidon}
if all its pairwise sums are different. In this entry point into the subject, combining
two elementary proofs, we decrease the gap between the upper and lower bounds by \(0.2\%\)
for infinitely many values of \(n\). We show that the maximum size of a Sidon set
of \(\{ 1, 2, \ldots, n\}\) is at most \(\sqrt{n}+ 0.998n^{1/4}\) for \(n\) sufficiently
large. Joint work with Zoltán Füredi and Souktik Roy.

**Ryan Martin, **Iowa State University

- February 25
- 2:30pm

**Abstract:** 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 Győri, 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.

**Hua Wang**, Georgia Southern University

- February 18
- 2:30pm

**Abstract:** A composition of a given positive integer n is an ordered sequence of positive integers
with sum n. In n-color compositions a part k has one of k possible colors. Using spotted
tiling to represent such colored compositions we consider those with restrictions
on colors. With general results on the enumeration of color restricted n-color compositions
in terms of allowed or prohibited colors, we introduce many particular combinatorial
observations related to various integer sequences and identities. This is joint work
with Brian Hopkins.

**Michael Levet**, University of Colorado Boulder

- February 11
- 2:30pm

**Abstract**:

The Weisfeiler–Leman (WL) algorithm is a key combinatorial subroutine in Graph Isomorphism, that (for fixed \(k \geq 2\)) computes an isomorphism invariant coloring of the k-tuples of vertices. Brachter & Schweitzer (LICS 2020) recently adapted WL to the setting of groups. Using a classical Ehrenfeucht–Fraïssé pebble game, we will show that Weisfeiler–Leman serves as a polynomial-time isomorphism test for several families of groups previously shown to be in \(\textsf{P}\) by multiple methods. These families of groups include:

- Coprime extensions \(H \ltimes N\), where \(H\) is \(O(1)\)-generated and the normal Hall subgroup \(N\) is Abelian (Qiao, Sarma, & Tang, STACS 2011).
- Groups without Abelian normal subgroups (Babai, Codenotti, & Qiao, ICALP 2012).

In both of these cases, the previous strategy involved identifying key group-theoretic structure that could then be leveraged algorithmically, resulting in different algorithms for each family. A common theme among these is that the group-theoretic structure is mostly about the action of one group on another. Our main contribution is to show that a single, combinatorial algorithm (Weisfeiler–Leman) can identify those same group-theoretic structures in polynomial time.

We also show that Weisfeiler–Leman requires only a constant number of rounds to identify groups from each of these families. Combining this result with the parallel WL implementation due to Grohe & Verbitsky (ICALP 2006), this improves the upper bound for isomorphism testing in each of these families from \(\textsf{P}\) to \(\textsf{TC}^0\).

This is joint work with Joshua A. Grochow.

**Stephan Wagner**, Uppsala University

- February 4
- 2:30pm

**Abstract**: A subtree of a graph is any (not necessarily induced) subgraph that is also a tree.
In this talk, I will discuss different questions concerning the distribution of subtrees
in deterministic and random graphs. For instance: what can we say about the distribution
of the subtree sizes, or the average size? What is the probability that a random subtree
is spanning (covers all vertices)?

**Ervin Gyõri**, Alfréd Rényi Institute of Mathematics

- January 28
- 2:30pm

**Abstract**: Let \(\text{ex}_P(n, H)\) denote the maximum number of edges in an \(n\)-vertex planar
graph which does not contain \(H\) as a subgraph. The topic of extremal planar graphs
was initiated by Dowden (2016). He obtained sharp upper bound for both \(\text{ex}_P(n,
C_4)\) and \(\text{ex}_P(n, C_5)\). Later on, Y. Lan, et al. continued this topic
and proved that \(\text{ex}_P(n, C_6) \le 18(n-2)/7\). In this talk, I will focus
on a sharp upper bound \(\text{ex}_P(n, C_6) \le 5n/2 - 7\), for all \(n \ge 18\),
which improves Lan’s result. Some related extremal planar graph problems will be discussed
shortly too.

This is joint result with Debarun Ghosh, Ryan R. Martin, Addisu Paulos, and Chuanqi Xiao.

**Youngho Yoo**, Georgia Tech

- December 3
- 2:30pm

**Abstract: **We show that if \(G\) is a 2-connected simple subcubic graph on \(n\) vertices with
\(n_2\) vertices of degree 2, then \(G\) has a TSP walk (spanning closed walk) of
length at most \(\frac{5n+n_2}{4}-1\), confirming a conjecture of Dvořák, Kráľ, and Mohar. This upper bound is best possible; there are infinitely many subcubic
(respectively, cubic) graphs whose minimum TSP walks have lengths \(\frac{5n+n_2}{4}-1\)
(respectively, \(\frac{5n}{4} - 2\)). Our proof implies a polynomial-time algorithm
for finding such a TSP walk. In particular, we obtain a \(\frac{5}{4}\)-approximation
algorithm for the graphic TSP on simple cubic graphs, improving on the previously
best known approximation ratio of \(\frac{9}{7}\).

Joint work with Michael Wigal and Xingxing Yu.

**He Guo**, Technion - Israel Institute of Technology

- November 19
- 2:30pm

**Abstract**: The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the
1970s. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension
of the binomial random graph is typically of order n/log n for constant edge-probabilities.
The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for
random hypergraphs with large uniformities, *i.e.*, edges of size O(log n).

Based on joint work with Kalen Patton and Lutz Warnke.

**Xiaonan Liu**, Georgia Tech

- November 12
- 2:30pm

**Abstract**: Hakimi, Schmeichel, and Thomassen conjectured that every \(4\)-connected planar
triangulation \(G\) on \(n\) vertices has at least \(2(n-2)(n-4)\) Hamiltonian cycles,
with equality if and only if \(G\) is a double wheel. In this paper, we show that
every \(4\)-connected planar triangulation on \(n\) vertices has \(\Omega(n^2)\) Hamiltonian
cycles. Moreover, we show that if \(G\) is a \(4\)-connected planar triangulation
on \(n\) vertices and the distance between any two vertices of degree \(4\) in \(G\)
is at least \(3\), then \(G\) has \(2^{\Omega(n^{1/4})}\) Hamiltonian cycles. Joint
work with Zhiyu Wang and Xingxing Yu.

**Grant Fickes**, Department of Mathematics, University of South Carolina

- November 5
- 2:30pm

**Abstract**: Call a hypergraph \(k\)-uniform if every edge is a \(k\)-subset of the vertices.
Moreover, a \(k\)-uniform linear hyperpath with \(n\) edges is a hypergraph \(G\)
with edges \(e_1, ... , e_n\) so that \(e_i \cap e_j = \emptyset\) if \(|i-j| \neq
1\), and \(|e_i \cap e_{i+1}| = 1\) for all \(1 \leq i\leq n-1\). Since the adjacency
matrix of a graph is real symmetric, sets of eigenvectors form vector spaces. The
extension of adjacency matrices to adjacency tensors in the hypergraph setting implies
eigenvectors are no longer closed under arbitrary linear combinations. Instead, eigenvectors
form varieties, and there is interest in finding decompositions of eigenvarieties
into irreducible components. In this talk we find such a decomposition of the nullvariety
for 3-uniform hyperpaths of arbitrary length. Moreover, we explore the repercussions
of this decomposition on a conjecture of Hu and Ye.

**Clifford Smyth**, Department of Mathematics and Statistics, University of North Carolina, Greensboro

- October 29
- 2:30pm

**Abstract: **A bond of a graph G is a spanning subgraph H such that each component of H is an induced
subgraph of G. The bonds of a graph G, when ordered by inclusion, form the so-called
bond lattice, L(G) of G, an interesting sub-lattice of the well-studied set partition
lattice. Bond lattices enjoy many interesting algebraic properties and there are interesting
combinatorial formulations of the Moebius function and characteristic polynomial of
these lattices.

We order the vertices of G and say two components in a bond are crossing if there are edges ik in one and jl in the other such that i < j < k < l. We say a bond is non-crossing if no two of its components cross. We study the non-crossing bond poset, NC(G), the set of non-crossing bonds of G ordered by inclusion, an interesting sub-poset of the well-studied non-crossing set partition lattice. We study to what extent the combinatorial theorems on L(G) have analogues in NC(G). This is joint work with Matt Farmer and Joshua Hallam.

**Anton Bernshteyn**, Department of Mathematics, Georgia Institute of Technology

- October 22
- 2:30pm

**Abstract**: According to a celebrated theorem of Johansson, every triangle-free graph \(G\)
of maximum degree \(\Delta\) has chromatic number \(O(\Delta/\log\Delta)\). Molloy
recently improved this upper bound to \((1+o(1))\Delta/\log\Delta\). In this talk,
we will strengthen Molloy’s result by establishing an optimal lower bound on the number
of proper \(q\)-colorings of \(G\) when \(q\) is at least \((1+o(1))\Delta/\log\Delta\).
One of the main ingredients in our argument is a novel proof technique introduced
by Rosenfeld. This is joint work with Tyler Brazelton, Ruijia Cao, and Akum Kang.

**Zhiyu Wang**, Mathematics, Georgia Institute of Technology

- October 15
- 2:30pm

**Abstract**: A graph class is called polynomially \(\chi\)-bounded if there is a function \(f\)
such that \(\chi(G) \leq f(\omega(G))\) for every graph \(G\) in this class, where
\(\chi(G)\) and \(\omega(G)\) denote the chromatic number and clique number of \(G\)
respectively. A *t-broom* is a graph obtained from \(K_{1,t+1}\) by subdividing an edge once. A *fork* is a graph obtained from \(K_{1,4}\) by subdividing two edges. We show two conjectures:
(1) we show that for graphs \(G\) without induced \(t\)-brooms, \(\chi(G) = o(\omega(G)^{t+1})\),
answering a question of Schiermeyer and Randerath. For \(t=2\), we strengthen the
bound on \(\chi(G)\) to \(7.5\omega(G)^2\), confirming a conjecture of Sivaraman.
(2) We show that any \{triangle, fork\}-free graph \(G\) satisfies \(\chi(G)\leq \omega(G)+1\),
confirming a conjecture of Randerath.

**Xiaofan Yuan**, Mathematics, Georgia Tech

- October 1
- 2:30pm

**Abstract**: The well-known Four Color Theorem states that graphs containing no *K*_{5}-subdivision or *K*_{3,3}-subdivision are 4-colorable. It was conjectured by Hajós that graphs containing no
*K*_{5}-subdivision are also 4-colorable. Previous results show that any possible minimum
counterexample to Hajós' conjecture is 4-connected but not 5-connected. We show that
any such counterexample does not admit a 4-cut with a nontrivial planar side. This
is joint work with Qiqin Xie, Shijie Xie and Xingxing Yu.

**Felix Lazebnik**, Department of Mathematical Sciences, University of Delaware

- September 24
- 2:30pm

**Abstract**: In this talk I will survey some results and open questions related to graphs of
high density and without cycles of length 4, or 6, or 8. Some of them are obtained
in a uniform way by lifting large induced subgraphs of Levi graphs (point-incidence
graphs) of projective planes.

Part of the talk will be based on recent joint work with Vladislav Taranchuk.

**Justin Troyka**, Mathematics, Davidson College

- September 17
- 2:30 pm

**Abstract**: A split graph is a graph whose vertices can be partitioned into a clique (complete
graph) and a stable set (independent set). How many split graphs on n vertices are
there? Approximately how many are there, as n goes to infinity? Collins and Trenk
(2018) have worked on these questions; after briefly summarizing their results, I
will show how I have generalized them in the setting of species theory, a powerful
framework for counting combinatorial objects acted on by isomorphisms. This generalization
leads to a result relating split graphs and bicolored graphs, allowing me to prove
the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are
"balanced".

**Joshua Cooper**, Department of Mathematics, University of South Carolina

- September 10
- 2:30pm

**Abstract**: Consider permutations \(\sigma : [n] \rightarrow [n]\) written in one-line notation
as a vector \(\vec{\sigma} = (\sigma(1),\ldots,\sigma(n))\). The permutohedron \(\mathcal{P}_n\)
(in one standard presentation) is the convex hull of all such \(\vec{\sigma}\), with
center \(\vec{p} = ((n+1)/2,\ldots,(n+1)/2)\). Then \(\mathcal{P}_n\) has a geometric
equator: permutations \(\tau\) so that \((\vec{\tau}-\vec{p}) \cdot (\vec{\text{id}}_n-\vec{p})
= 0\), where \(\text{id}_n\) is the identity permutation. But \(\mathcal{P}_n\) also
has a combinatorial equator: permutations \(\tau\) in the middle level of the weak
Bruhat order, i.e., for which \(\text{inv}(\tau) = \frac{1}{2} \binom{n}{2}\). We
ask: how close are these two equators to each other? This question arose in connection
with a problem in machine learning concerned with estimating so-called Shapley values
by sampling families of permutations efficiently.

The most interesting special case of our main result is that, for permutations \(\tau\) close to the geometric equator, i.e., for which \((\vec{\tau}-\vec{p}) \cdot (\vec{\text{id}}_n-\vec{p}) = o(1)\), the number of inversions satisfies

$$

\left | \frac{\text{inv}(\tau)}{\binom{n}{2}} - \frac{1}{2} \right | \leq \frac{1}{4}
+ o(1)

$$

and, furthermore, the quantity \(1/4\) above cannot be improved beyond \(1/2 - 2^{-5/3} \approx 0.185\). The proof is not difficult but uses an amusing application of bubble sorting. We discuss this and a more general and precise version of the result for any ratio \(\text{inv}(\tau)/\binom{n}{2}\).

Joint work with: Rory Mitchell of Nvidia, and Eibe Frank and Geoffrey Holmes of University of Waikato, NZ

**Andrew Meier**, Department of Mathematics, University of South Carolina

- September 3
- 2:30pm

**Abstract**: An edge-colored graph \(H\) is called \textit{rainbow} if every edge of \(H\) receives
a different color. Given any host multigraph \(G\), the *anti-Ramsey* number of \(t\) edge-disjoint rainbow spanning trees in \(G\), denoted by \(r(G,t)\),
is defined as the maximum number of colors in an edge-coloring of \(G\) containing
no \(t\) edge-disjoint rainbow spanning trees. For any vertex partition \(P\), let
\(E(P,G)\) be the set of non-crossing edges in \(G\) with respect to \(P\). In this
talk, we determine \(r(G,t)\) for all host multigraphs \(G\): \(r(G,t)=|E(G)|\) if
there exists a partition \(P_0\) with \(|E(G)|-|E(P_0,G)|<t(|P_0|-1)\); and \(r(G,t)=\max_{P\colon
|P|\geq 3} \{|E(P,G)|+t(|P|-2)\}\) otherwise. As a corollary, we determine \(r(K_{p,q},t)\)
for all values of \(p,q, t\), improving a result of Jia, Lu and Zhang.