### 2021–2022 Academic Year

**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.