### 2024 – 2025 Academic Year

**Organized by:** George Brooks (ghbrooks@email.sc.edu) and Utku Okur (uokur@email.sc.edu)

This page will be updated as new seminars are scheduled. Make sure to check back each week for information on upcoming seminars.

Many of the presentations of this seminar are recorded with the speaker's permission. These recordings can be found on the UofSC Discrete Mathematics Seminar YouTube channel.

An email is sent out each week with specifics about the upcoming talk which includes information about format, time, place, etc. If you would like to receive these emails or have other questions, please contact George Brooks or Utku Okur.

**When:** Friday, November 22^{nd}, 2024

**Speaker:** Svetlana Poznanovikj

**Location:** TBA

**Abstract: **TBA

**When:** Friday, November 15^{th}, 2024

**Speaker:** Josh Cooper

**Location:** TBA

**Abstract: **TBA

**When:** Friday, November 8^{th}, 2024

**Speaker:** Alec Helm

**Location:** TBA

**Abstract: **TBA

**When:** Friday, November 1^{st}, 2024

**Speaker:** Vladislav Taranchuk

**Location:** TBA

**Abstract: **TBA

**When:** Friday, October 25^{th}, 2024

**Speaker:** Himanshu Gupta

**Location:** TBA

**Abstract: **TBA

**When:** Friday, October 11^{th}, 2024

**Speaker:** Josiah Reiswig

**Location:** TBA

**Abstract: **TBA

**When:** Friday, September 13^{th}, 2024

**Speaker:** Sam Spiro

**Location:** TBA

**Abstract: **TBA

### Previous Years

**When:** April 19^{th} at 2:30 p.m.

**Speaker:** Jonad Pulaj (Davidson College)

**Location:** Virtual via Zoom

**Abstract: **In the first part of this talk we give a brief overview of computer-assisted results
in combinatorics, before focusing on building safe computational frameworks for Frankl's
conjecture combining mixed integer and linear programming (MILP) with satisfiability
modulo theories (SMT). The conjecture states that for any finite union-closed family
of sets which contains a non-empty set, there exists an element in the union of sets
of the family that is present in at least half the sets in the family. Our framework
facilitates and/or inspires various results including the following. We find new values
and bounds for the least integer m such that any family containing m distinct k-sets
of an n-set X satisfies Frankl's conjecture with an element of X. Furthermore, we
settle an older question of Vaughan about symmetry in union-closed families and we
prove a recent question posed by Ellis, Ivan and Leader.

In the last part of this talk we focus on general best practices and desirable standards for computer-assisted results, before focusing on the verification of answers obtained from MILP certificates using SMT solvers. We present an SMT-based tool that checks the VIPR certificate, the first proof format for the correctness of answers produced MILP solvers. This checker is based on the equivalence between the correctness of a VIPR certificate and the satisfiability of a formula in the theory of linear/integer real arithmetic. Finally we demonstrate the effectiveness of this approach and its variations by evaluating them on existing benchmark instances.

Collaborators in these projects include Hammurabi Mendes, Kenan Wood, Haoze (Andrew) Wu, and Runtian (Daniel) Zhou.

**When:** April 12^{th} at 2:30 p.m.

**Speaker: **Michael Levet (College of Charleston)

**Location: **LeConte 103

**Abstract: ** The *Graph Isomorphism* problem takes as input two finite graphs \(G\) and \(H\), and asks if \(G \cong H\).
It is a longstanding open problem whether *Graph Isomorphism* is solvable in polynomial-time. In this talk, we will discuss recent advances in
the computational complexity of identifying graphs of bounded rank-width, a class
of dense graphs for which isomorphism testing has only recently been shown to belong
to \(\textsf{P}\). Precisely, we will show that \(O(\log n)\) rounds of the constant-dimensional
Weisfeiler--Leman algorithm serves as a complete isomorphism test for this family.
This, in particular, improves the complexity-theoretic upper bound from \(\textsf{P}\)
to \(\textsf{TC}^{1}\). We then leverage the Weisfeiler--Leman algorithm as a subroutine,
to obtain a \(\textsf{TC}^{2}\) algorithm to compute canonical labelings for this
family.

This is joint work with Puck Rombach and Nicholas Sieger.

**When: **April 5^{th} at 2:30 p.m.

**Speaker: **Maria-Romina Ivan (University of Cambridge)

**Location: **Virtual via Zoom

**Abstract: **The Turán density of an \(r\)-uniform hypergraph \(H\), denoted by \(\pi(H)\), is
the limit of the maximum density of an \(n\)-vertex \(r\)-uniform hypergraph not containing
a copy of \(H\), as \(n\) tends to infinity. An \(r\)-daisy is an \(r\)-uniform hypergraph
consisting of the six \(r\)-sets formed by taking the union of an \((r-2)\)-set with
each of the \(2\)-sets of a disjoint \(4\)-set. Bollobás, Leader and Malvenuto, and
also Bukh, conjectured that the Turán density of the \(r\)-daisy is zero. A folklore
Turán-type conjecture for hypercubes states that for fixed \(d\) the smallest set
of vertices of the \(n\)-dimensional hypercube \(Q_n\) that meets every copy of \(Q_d\)
has density \(1/(d+1)\) as \(n\) goes to infinity. In this talk, we show that the
Turán density for daisies is positive, and, by adapting our construction, we also
disprove the hypercube conjecture mentioned above. Joint work with David Ellis and
Imre Leader.

**When: **March 29^{th} at 2:30 p.m.

**Speaker:** Zhiyu Wang (Louisiana State University)

**Location:** Virtual via Zoom

**Abstract: **The oriented diameter of an undirected graph \(G\) is the minimum diameter of an orientation
of \(G\) over all strongly connected orientations of \(G\). In this talk, we will
discuss some recent progress on the oriented diameter of some graph classes including
connected graphs with minimum degree \(\delta\) and planar triangulations.

**When:** March 22^{nd} at 2:30 p.m.

**Speaker: **Megan Cream (Lehigh University)

**Location:** Virtual via Zoom

**Abstract:** This talk explores various conditions that imply certain cycle properties in graphs.
A graph G of order n is called pancyclic if it contains a cycle of every possible
length from 3 to n, and a cycle C is chorded if there is an edge between two vertices
that are non-adjacent on C. In this talk, we define multiple relaxations of pancyclic
graphs and we extend those cycle properties to chorded cycle properties. In 1971,
Bondy proposed a meta-conjecture stating that any condition implying hamiltonicity
in a graph will also imply pancyclicity. In 2017, we extended this meta-conjecture
to say any condition implying hamiltonicity will also imply chorded pancyclicity.
The results discussed in this talk are evidence that support this new meta-conjecture.

**When: **March 15^{th} at 2:30 p.m.

**Speaker: **Lina Li (Iowa State University)

**Location: **Virtual via Zoom

**Abstract: **The Acyclic Edge Coloring Conjecture, posed independently by Fiamčik in 1978 and Alon,
Sudakov and Zaks in 2001, states that every graph can be properly edge colored with
\(\Delta+2\) colors such that there is no bicolored cycle. Over the years, this conjecture
has attracted much attention. We prove that the conjecture holds asymptotically, that
is \((1+o(1))\Delta\) colors suffice. This is joint work with Michelle Delcourt and
Luke Postle.

**When:** March 1^{st} at 2:30 p.m.

**Speaker:** Michael Tait (Villanova University)

**Location:** LeConte 103

**Abstract:** A subset of integers A is a \(B_k[g]\) set if the number of multisets of A of size
k which sum to any fixed integer is bounded above by g. How large can a \(B_k[g]\)
set of integers up to n be? We will discuss what is known about \(B_k[g]\) sets and
how these objects are related to extremal graph theory. This is joint work with Griffin
Johnston and Craig Timmons.

**When:** February 23^{rd} at 2:30 p.m.

**Speaker:** Stephan Wagner (Uppsala University)

**Location:** Virtual via Zoom

**Abstract:** We consider random digraphs in a directed version of the classical Erdős–Rényi model:
given n vertices, each possible directed edge is inserted with probability p, independently
of the others. It turns out that these graphs undergo a phase transition when p is
about 1/n, which can be seen in the answer to questions such as: what is the probability
that there are no directed cycles (equivalently, that all strongly connected components
are singletons)? Using methods from analytic combinatorics, we obtain very precise
asymptotic answers to questions of this kind.

**When:** February 16^{th} at 2:30 p.m.

**Speaker:** Nika Salia (King Fahd University)

**Location:** Virtual via Zoom

**Abstract:** In our talk, we study embeddings of planar graphs on a plane and explore the limits
of deviating from optimal behavior. We introduce the concept of a plane-saturated
subgraph for a planar graph, defined as a subgraph where adding any edge would compromise
planarity or no longer maintain its status as a subgraph. Our focus lies on quantifying
this phenomenon through the plane-saturation ratio, denoted as \(psr(G)\), which measures
the minimum ratio of edges in a plane-saturated subgraph to the total edges in the
original graph \(G\). While some planar graphs have arbitrarily small \(psr(G)\),
we show that for all twin-free planar graphs, \(psr(G)>1/16\), and that there exist
twin-free planar graphs where \(psr(G)\) is arbitrarily close to \(1/16\).

Joint work with Dr. Alexander Clifton, Institute for Basic Science.

**When:** February 9^{th} at 2:30 p.m.

**Speaker: ** George Brooks (University of South Carolina)

**Location:** LeConte 103

**Abstract:** In differential geometry, the Ricci curvature of a Riemannian manifold is, roughly
speaking, a measure of how much the manifold differs from Euclidean space. Curvature
in the continuous setting has been studied in depth throughout the past 200 years,
and more recent interest has arisen in establishing analogous results in the discrete
setting. While Riemannian manifolds generally behave quite differently than graphs,
we can use random walks to define the Ricci curvature on graphs. In this talk, we'll
review Lin-Lu-Yau curvature and consider the question: Can we determine an upper bound
on the order for all positively curved graphs in graph class?

**When: **February 2^{nd} at 2:30 p.m.

**Speaker:** Utku Okur (University of South Carolina)

**Location:** LeConte 103

**Abstract:** We show that, in general, the characteristic polynomial of a hypergraph is not determined
by its "polynomial deck", the multiset of characteristic polynomials of its vertex-deleted
subgraphs, thus settling the "polynomial reconstruction problem" for hypergraphs in
the negative. The proof proceeds by showing that a construction due to Kocay of an
infinite family of pairs of 3-uniform hypergraphs which are non-isomorphic but share
the same hypergraph deck, in fact, have different characteristic polynomials. The
question remain unresolved for ordinary graphs.

**When:** January 26^{th} at 2:30 p.m.

**Speaker:** Carl Yerger (Davidson College)

**Location:** LeConte 103

**Abstract:** Graph pebbling is a combinatorial game played on an undirected graph with an initial
configuration of pebbles. A pebbling move consists of removing two pebbles from one
vertex and placing one pebbling on an adjacent vertex. The pebbling number of a graph
is the smallest number of pebbles necessary such that, given any initial configuration
of pebbles, at least one pebble can be moved to a specified target vertex. In this
talk, we will give a survey of several streams of research in pebbling, including
describing a theoretical and computational framework that uses mixed-integer linear
programming to obtain bounds for the pebbling numbers of graphs. We will also discuss
improvements to this framework through the use of newly proved weight functions that
strengthen the weight function technique of Hurlbert. Finally, we will discuss some
open extremal problems in pebbling, specifically related to Class 0 graphs and describe
how structural graph theoretic techniques such as discharging can be used to obtain
results.

Collaborators on these projects include Dan Cranson, Dominic Flocco, Luke Postle, Jonad Pulaj, Chenxiao Xue, Marshall Yang, Daniel Zhou.

**When:** January 19^{th} at 2:30 p.m.

**Speaker:** Tom Kelly (Georgia Tech)

**Location:** LeConte 346

**Abstract: **In this talk, we will discuss *robustness* results which lie in the intersection of both extremal and probabilistic combinatorics.

In joint work with Kang, Kühn, Methuku, and Osthus, we proved the following: If \(p\geq{C\log^2n/n}\) and \(L_{i,j}\subseteq[n]\) is a random subset of \([n]\) where each \(k\in[n]\) is included in \(L_{i,j}\) independently with probability \(p\) for each \(i,j\in[n]\), then asymptotically almost surely there is an order-\(n\) Latin square in which the entry in the \(i\)th row and \(j\)th column lies in \(L_{i,j}\). We prove analogous results for Steiner triple systems and \(1\)-factorizations of complete graphs. These results can be understood as stating that these "design-like" structures exist "robustly".

In joint work with Kang, Küuhn, Osthus, and Pfenninger, we proved various results stating that if \(\mathcal{H}\) is an \(n\)-vertex \(k\)-uniform hypergraph satisfying some minimum degree condition and \(p=\Omega(n^{-k+1}\log n)\), then asymptotically almost surely a \(p\)-random subhypergraph of \(\mathcal{H}\) contains a perfect matching. These results can be understood as "robust" versions of hypergraph Dirac-type results as they simultaneously strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem on the threshold for when a binomial random \(k\)-uniform hypergraph contains a perfect matching. In joint work with Müyesser, and Pokrovskiy, we proved similar results for hypergraph Hamilton cycles.

All of these results utilize the recent Park—Pham Theorem or one of its variants.
A crucial notion for this is that of the *spreadness* of a certain type of probability distribution.

**When:** January 12^{th} at 2:30 p.m.

**Speaker: ** Leslie Hogben (Iowa State University)

**Location:** Virtual via Zoom

**Abstract: **Zero forcing is an iterative process that repeatedly applies a rule to change the
color of vertices of a graph *G* from white to blue. The zero forcing number is the minimum number of initially blue
vertices that are needed to color all vertices blue through this process. Standard
zero forcing was introduced about fifteen years ago in the control of quantum systems
and as an upper bound for maximum multiplicity of an eigenvalue (or maximum nullity)
among matrices having off-diagonal nonzero pattern described by the edges of the graph
*G*, and rediscovered later both as part of power domination and as fast-mixed graph
searching.

Whether a set is a zero forcing set can be tested using a certain type of set called
a fort, which obstructs zero forcing. The maximum number of disjoint forts (fort
number) provides another lower bound for the zero forcing number; results about
fort number will be discussed. Forts can be used in integer programs to determine
the zero forcing number and fort number. The relaxation of these integer programs
leads to dual linear programs that define the fractional zero forcing number, or equivalently,
the fractional fort number, and results about these parameters will be discussed.

There is a well-known upper bound for the zero forcing number of a Cartesian product
in terms of the zero forcing numbers and orders of the constituent graphs. The question
of a lower bound for the zero forcing number of a Cartesian product has recently been
studied. It is easy to see that there is a Vizing-like lower bound when the constituent
graphs of the Cartesian product both have maximum nullity equal to zero forcing number.
Fractional zero forcing and fort number provide additional lower bounds on the the
zero forcing number of a Cartesian product in terms of parameters of the constituent
graphs.

**When:** December 8^{th} at 2:30 p.m.

**Speaker:** Gabor Hetyei (UNC Charlotte)

**Location:** LeConte 103

**Abstract:** The tensor product of a graph and of a pointed graph is obtained by replacing each
edge of the first graph with a copy of the second. In this expository talk we will
explore a colored generalization of Brylawski's formula for the Tutte polynomial
of the tensor product of a graph with a pointed graph and its applications. Using
Tutte's original (activity-based) definition of the Tutte polynomial we will provide
a simple proof of Brylawski's formula. This can be easily generalized to the colored
Tutte polynomials introduced by Bollobas and Riordan. Consequences include formulas
for Jones polynomials of (virtual) knots and for invariants of composite networks
in which some major links are identical subnetworks in themselves.

All results presented are joint work with Yuanan Diao, some of them are also joint
work with Kenneth Hinson. The relevant definitions and the fundamental results used
will be carefully explained.

**When:** December 1^{st} at 2:30 p.m.

**Speaker:** Miles Bakenhus (Illinois Institute of Technology)

**Location:** Virtual via Zoom

**Abstract:** The set of nonnegative integer lattice points in a polytope, also known as the fiber
of a linear map, makes an appearance in several applications including optimization
and statistics. We address the problem of sampling from this set using three ingredients:
an easy-to-compute lattice basis of the constraint matrix, a biased sampling algorithm
with a Bayesian framework, and a step-wise selection method. The bias embedded in
our algorithm updates sampler parameters to improve fiber discovery rate at each step
chosen from previously discovered elements. We showcase the performance of the algorithm
on several examples, including fibers that are out of reach for the state-of-the-art
Markov bases samplers. Potential applications in integer optimization are considered
as well.

**When:** November 17^{th} at 2:30 p.m

**Speaker: **David Galvin (University of Notre Dame)

**Location:** LeConte 103

**Abstract:** In the late 1980's, motivated by a problem from combinatorial group theory, Andrew
Granville asked "at most how many independent sets can a regular graph admit?" This
has proven to be a remarkably fruitful question, and one amenable to a remarkable
variety of techniques, including graph containers, entropy and linear programming.

Although Granville's initial question has by now been completely resolved (first by
Kahn, for regular bipartite graphs, then by Zhao for all regular graphs), many related
open questions remain.

I'll discuss this problem, and some of its relatives, including maximizing the count
of H-colorings of a regular graph, minimizing the count of H-colorings of a tree,
and maximizing the count of independent sets in a regular graph in the presence of
a bound on the largest independent set in the graph.

**When:** November 10^{th} at 2:30pm

**Speaker:** Nicholas Sieger (University of California; San Diego)

**Location:** Virtual via Zoom

**Abstract:** We introduce a random graph model for clustering graphs with a given degree sequence.
Unlike many previous random graph models, we incorporate clustering effects into the
model. We show that random clustering graphs can construct graphs with a power-law
expected degree sequence, small diameter, and any desired clustering coefficient.
Our results follow from a general theorem on subgraph counts which may be of independent
interest.

**When:** November 3^{rd} at 2:30pm

**Speaker: **Clifford Smyth (UNC Greensboro)

**Location:** Virtual via Zoom

**Abstract: **The reciprocal of \(e^{-x}\) is \(e^x\) and \(e^{x} = sum_{n=0}^\infty x^n/n!\) is
"totally non-negative" in the sense that it has a power series with all non-negative
coefficients.

What about "thinned versions" of \(e^{-x} = 1 + \sum_{n=1}^\infty (-1)^n x^n/n!\),
i.e. functions of the form \(f_A(x) = 1 + \sum{n \in A} (-1)^n x^n/n!\) where \(A\)
is some subset of the positive integers? For which \(A\) does \(f_A\) have a totally
non-negative reciprocal? Gessel showed that \(A = \{1,2,...,2m\}\) does not work,
but \(A = \{0,1,...,2m+1\}\) does. In fact, he showed that in the latter case, \(1/f_A(x)
= \sum_n c(n) x^n/n!\) where \(c(n)\) counts the number of elements in class \(C(n)\)
of combinatorially defined objects. \(C(n)\) turns out to be the permutations of
\(\{1,...,n\}\) in which the length of every maximal increasing subsequence is congruent
to 0 or 1 mod 2m. Thus, Gessel showed that \(1/f_A(x)\) is totally non-negative for
a "combinatorially interesting" reason when \(A = \{1,...,2m+1\}\).

We extend this result by showing that \(1/f_A(x)\) is totally non-negative for any
\(A\) that contains 1 and is "odd-ended". By odd-ended, we mean that \(A\) has all
of its maximal intervals of consecutive members ending in odd integers, like \(A =
\{1,2,3\} \cup \{5,6,7\} \cup \{11\} \cup \{15,16,...., \infty\}\). We too give combinatorially
interesting reasons for this. For each such \(A\), we show that \(1/f_A(x) = \sum_n
c(A,n) x^n/n!\) where \(c(A,n)\) counts the number of elements in a class \(C(A,n)\)
of combinatorially defined objects. We push things a little further than that as
well, finding a lot more "good" \(A\)'s than the odd-ended ones.

This is joint work with David Galvin of the University of Notre Dame and John Engbers
of Marquette University.

**When:** October 27^{th} at 2:30pm

**Speaker:** Stacie Baumann (College of Charleston)

**Location:** Virtual via Zoom

**Abstract:** An (n,k,t)-graph is a graph on n vertices in which every set of *k *vertices contain a clique on *t* vertices. Turán's Theorem (complemented) states that the unique minimum (n,k,2)-graph
is a disjoint union of cliques. We prove that minimum (n,k,t)-graphs are always disjoint
unions of cliques for any *t* (despite nonuniqueness of extremal examples), thereby generalizing Turán's Theorem
and confirming two conjectures of Hoffman *et al*. This is joint work with Joseph Briggs.

**When: **October 6^{th} at 2:30pm

**Speaker: **Ryan Martin (Iowa State University)

**Abstract: **Basic Turan theory asks how many edges a graph can have, given certain restrictions
such as not having a large clique. A more generalized Turan question asks how many
copies of a fixed subgraph \(H\) the graph can have, given certain restrictions. There
has been a great deal of recent interest in the case where the restriction is planarity.
In this talk, we will discuss some of the general results in the field, primarily
the asymptotic value of \(bf N_{\mathcal P}(n,H)\), which denotes the maximum number
of copies of \(H\) in an \(n\)-vertex planar graph. In particular, we will focus on
the case where \(H\) is a cycle. It was determined that \(bf N_{\mathcal P}(n,C_{2m})
= (n/m)^m + o(n^m)\) for small values of \(m\) by Cox and Martin and resolved for
all \(m\) by Lv, Gy\H{o}ri, He, Salia, Tompkins, and Zhu. The case of \(H=C_{2m+1}\)
is more difficult and it is conjectured that \(bf N_{\mathcal P}(n,C_{2m+1})=2m(n/m)^m+o(n^m)\).
We will duscuss recent progress on this problem, including verification of the conjecture
in the case where \(m=3\) and \(m=4\) and a lemma which reduces the solution of this
problem for any \(m\) to a so-called ''maximum likelihood'' problem. The maximum likelihood
problem is, in and of itself, an interesting question in random graph theory. This
is joint work with Emily Heath and Chris (Cox) Wells.

**When: **September 29^{th} at 2:30pm

**Speaker: **Grace McCourt (University of Illinois at Urbana-Champaign)

**Abstract: **Dirac proved that each *n*-vertex 2-connected graph with minimum degree at least *k *contains a cycle of length at least min{2k, n}. We consider a hypergraph version of
this result. A Berge cycle in a hypergraph is an alternating sequence of distinct
vertices and edges v_{1},e_{2},v_{2}, ... , e_{c}, v_{1} such that {v_{i},v_{i+1}} ⊆ e_{i} for all *i *(with indices taken modulo *c*). We prove that for n ≥ k ≥ r+2 ≥ 5, every 2-connected *r*-uniform *n*-vertex hypergraph with minimum degree at least {k-1 \choose r-1} + 1 has a Berge
cycle of length at least min{2k, n}. The bound is exact for all k ≥ r+2 ≥ 5. This is joint work with Alexandr Kostochka and Ruth Luo.

**When: September** 22^{nd} at 2:30pm

**Speaker: **Ruth Luo (University of South Carolina)

**Abstract: ** Let A and B be 0,1-matrices. We say A contains B as a configuration if there exists
a submatrix of A that is a row and column permutation of B. In the special case where
A and B are incidence matrices of graphs G and H respectively, A contains B as a configuration
if and only if B is a subgraph of A. We discuss some extremal problems where A and
B are incidence matrices of (hyper)graphs.

**When: **September 15^{th} 2023 at 2:30pm

**Speaker:** Sebastian Cioaba (University of Delaware)

**Abstract:** Rigidity is the property of a structure that does not flex. It is well studied in
discrete geometry and mechanics, and has applications in material sciences, engineering
and biological sciences. In 1982, Lovasz and Yemini proved that every 6-connected
graph is rigid. Consequently, if Fiedler’s algebraic connectivity is greater than
5, then G is rigid. In recent work with Sean Dewar and Xiaofeng Gu, we showed that
for a graph G with minimum degree d>5, if its algebraic connectivity is greater than
2+1/(d-1), then G is rigid and if its algebraic connectivity is greater than 2+2/(d-1),
then G is globally rigid. Our results imply that every connected regular Ramanujan
graph with degree at least 8 is globally rigid. Time permitting, I will report on
another recent work with Sean Dewar, Georg Grasegger and Xiaofeng Gu extending these
results to valencies smaller than 8.

**When: **September 8^{th} 2023 at 2:30pm

**Speaker: **Bruce Sagan (Michigan State University)

**Abstract: **Let *t *be a positive integer and let *G *be a combinatorial graph with vertices*V *and edges *E*. A proper coloring of *G *from a set with *t *colors is a function *c :**V → {1, 2, . . . , t}* such that if *uv ∈ E* then *c(u) ̸= c(v)*, that is, the endpoints

of an edge must be colored differently. These are the colorings considered in

the famous Four Color Theorem. The chromatic polynomial of *G*, *P(G;t),*

is the number of proper colorings of *G *from a set with *t *colors. It turns out

that this is a polynomial in *t *with many amazing properties. For example,

there are connections with acyclic orientations, hyperplane arrangements and

symmetric functions. This talk will survey some of these results.

**When: **September 1^{st} 2023 at 2:30pm

**Speaker: **William Linz (University of South Carolina)

**Abstract: ** For positive integers n and k, an L-system is a collection of k-uniform subsets of
a set of size n whose pairwise intersection sizes all lie in in the set L. The maximum
size of an L-system is equal to the independence number of a certain union of graphs
in the Johnson scheme. The Lovasz number is a semidefinite programming approximation
of the independence number of a graph. In this talk, we survey the relationship between
the maximum size of an L-system and the Lovasz number, illustrating examples both
where the Lovasz number is a good approximation and where it is a bad approximation.

**Ruth Luo**, University of South Carolina

- Sep 23
- 2:30pm

**Abstract**: A graph is hamiltonian-connected if for every pair of vertices u and v there exists
a hamiltonian path from u to v. It is easy to see that hamiltonian-connectness implies
hamiltonicity in graphs. In this talk, we consider r-uniform hypergraphs and give
sufficient conditions for the existence of hamiltonian Berge paths between any pair
of vertices. It is also shown that this property in hypergraphs does not necessarily
imply the existence of a hamiltonian Berge cycle.This is joint work with Alexandr
Kostochka and Grace McCourt

** **

**Himanshu Gupta**, University of Delaware

- Sep 23
- 2:30pm

**Abstract: **Embedding graphs into Euclidean spaces with least distortion is a topic well-studied
in mathematics and computer science. Despite this research, there are just a few graphs
for which the precise least distortion and a least distortion embedding is known.
In 2008, Vallentin studied this problem for distance-regular graphs and obtained a
lower bound for the least distortion of a distance-regular graph. In addition, he
showed that this bound is tight for Hamming and Johnson graphs as well as strongly
regular graphs and conjectured that his bound is always tight for distance-regular
graphs. In this talk, we provide several counterexamples to this conjecture with diameter
4 and larger, but we also prove the conjecture for several families of distance-regular
graphs. This is joint work with Sebastian M. Cioaba (University of Delaware), Ferdinand
Ihringer (Ghent University), and Hirotake Kurihara (Yamaguchi University).

**Joshua Cooper**, University of South Carolina

- Sep 16
- 2:30pm

**Abstract: **A problem that arose in computational phylogenetics led to study of so-called "successful
pressing sequences" on graphs, permutations of the vertices with a certain elimination-type
property. It turns out that the success of a given pressing sequence can be stated
in a number of interesting ways, one of which is a linear-algebraic condition on the
graph’s adjacency matrix over GF(2) that is formally almost identical to semi-definiteness
over \(\mathbb{R}\). We explore which classical properties of semi-definiteness carry
over to GF(2) and other finite fields, and which ones fail. Joint work with Erin
Hanna and Hays Whitlatch.

**William Linz**, University of South Carolina

- Sep 9
- 2:30pm

**Abstract**: A family of sets F is d-wise, t-intersecting if any d sets from F contain at least
t elements in common. In this talk, I will state and prove analogues of the classical
Erdős-Ko-Rado and Hilton-Milner theorems for d-wise, t-intersecting families, improving
upon several earlier results by a number of authors, including O'Neill-Verstraete
and Tokushige. I will also discuss various extensions of these results. Time permitting,
I will indicate how d-wise, t-intersecting families can be used to construct \(K_{s,t}\)-intersecting
families of size larger than the trivial construction, answering a question of Ellis.
This is joint work with Jozsef Balogh.

**Yan Zhaung**, Davidson College

- Sep 2
- 2:30pm

**Abstract**: The Goulden–Jackson cluster method, adapted to permutations by Elizalde and Noy,
reduces the problem of counting permutations by occurrences of a prescribed consecutive
pattern to that of counting *clusters*, which are special permutations with a lot of structure. Recently, the speaker proved
a lifting of the cluster method to the Malvenuto–Reutenauer algebra, which specializes
to Elizalde and Noy's cluster method for permutations as well as new formulas which
refine by additional permutation statistics, namely the inverse descent number ides,
the inverse peak number ipk, and the inverse left peak number ilpk. Thus, if σ is
a consecutive pattern and we can count σ-clusters by ides, ipk, or ilpk, then we can
count all permutations by occurrences of σ jointly with the corresponding statistic. This talk will survey results on counting clusters by the statistics ides, ipk, and
ilpk. Special attention will be given to the pattern 2134...*m*; in joint work with Sergi Elizalde and Justin Troyka, we show that 2134...(*r*+1)-clusters are equinumerous with *r*-*Stirling permutations* introduced by Gessel and Stanley, and we establish some joint equidistributions between
these two families of permutations.

** **

**Leslie Hogben**, Iowa State University and American Institute of Mathematics

- Aug 26
- 2:30pm

**Abstract**: Zero forcing is an iterative process that repeatedly applies a rule to fill vertices
of a graph. The zero forcing number is the minimum number of initially filled vertices
that are needed to fill all vertices through the process, and is denoted by Z(G).
The maximum multiplicity of an eigenvalue (or maximum nullity) among matrices having
off-diagonal nonzero pattern described by the edges of the graph G is denoted by M(G).
It is known that M(G) <= Z(G), and this relationship is one of the origins of zero
forcing (and the source of its name). Several variants of zero forcing number and
maximum nullity, such as positive semidefinite zero forcing number and maximum nullity,
have also been studied. For each variant of zero forcing, other parameters of interest
include the propagation time (number of rounds needed to fill all vertices) and throttling
number (which minimizes a combination of resources used to accomplish a task and rounds
needed to accomplish it). This talk will survey results on extreme values of various
zero forcing numbers, propagation times, throttling numbers, and maximum nullities.

**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.