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Series: Graph Theory Seminar

Since the foundational results of Thomason and
Chung-Graham-Wilson on quasirandom graphs over 20 years ago, there has
been a lot of effort by many researchers to extend the theory to
hypergraphs. I will present some of this history, and then describe our
recent results that provide such a generalization and unify much of the
previous work. One key new aspect in the theory is a systematic study of
hypergraph eigenvalues first introduced by Friedman and Wigderson. This
is joint work with John Lenz.

Series: Graph Theory Seminar

We show that any internally 4-connected non-planar bipartite graph contains
a subdivision of K3,3 in which each subdivided path contains an even number
of vertices. In addition to being natural, this result has broader
applications in matching theory: for example, finding such a subdivision of
K3,3 is the first step in an algorithm for determining whether or not a
bipartite graph is Pfaffian. This is joint work with Robin Thomas.

Series: Graph Theory Seminar

A recent lower bound on the number of edges in a k-critical n-vertex graph by
Kostochka and Yancey yields a half-page proof of the celebrated Grotzsch
Theorem that every planar triangle-free graph is 3-colorable. We use the same
bound to give short proofs of other known theorems on 3-coloring of planar
graphs, among whose is the Grunbaum-Aksenov Theorem that every planar with at
most three triangles is 3-colorable. We also prove the new result that every
graph obtained from a triangle-free planar graph by adding a vertex of degree
at most four is 3-colorable. Joint work with O. Borodin, A. Kostochka and M. Yancey.

Series: Graph Theory Seminar

We present a dynamic data structure representing a graph G, which allows
addition and removal of edges from G and can determine the number of
appearances of a graph of a bounded size as an induced subgraph of G. The
queries are answered in constant time. When the data structure is used to
represent graphs from a class with bounded expansion (which includes planar
graphs and more generally all proper classes closed on topological minors, as
well as many other natural classes of graphs with bounded average degree), the
amortized time complexity of updates is polylogarithmic. This data structure
is motivated by improving time complexity of graph coloring algorithms and
of random graph generation.

Series: Graph Theory Seminar

Every subcubic triangle-free graph on n vertices contains an independent
set of size at least 5n/14 (Staton'79). We strengthen this result by showing
that all such graphs have fractional chromatic number at most 14/5,
thus confirming a conjecture by Heckman and Thomas. (Joint work with J.-S. Sereni and J. Volec)

Series: Graph Theory Seminar

The online matching problem has received significant attention in recent years because of its connections to allocation problems in internet advertising, crowd sourcing, etc. In these real-world applications, the typical goal is not to maximize the number of allocations; rather it is to maximize the number of “successful” allocations, where success of an allocation is governed by a stochastic event that comes after the allocation. These applications motivate us to introduce stochastic rewards in the online matching problem. In this talk, I will formally define this problem, point out its connections to previously studied allocation problems, give a deterministic algorithm that is close to optimal in its competitive ratio, and describe some directions of future research in this line of work. (Based on joint work with Aranyak Mehta.)

Series: Graph Theory Seminar

The Weak Structure Theorem of Robertson and Seymour is the cornerstone of
many of the algorithmic applications of graph minors techniques. The
theorem states that any graph which has both large tree-width and excludes
a fixed size clique minor contains a large, nearly planar subgraph. In
this talk, we will discuss a new proof of this result which is
significantly simpler than the original proof of Robertson and Seymour. As
a testament to the simplicity of the proof, one can extract explicit
constants to the bounds given in the theorem. We will assume no previous
knowledge about graph minors or tree-width.
This is joint work with Ken Kawarabayashi and Robin Thomas

Series: Graph Theory Seminar

A (5,2)-configuration in a graph G is a function which maps the
vertices of G into 2-element subsets of {1,2,3,4,5} in such a way that
for every vertex u, the union of the 2-element subsets assigned to u and
all its neighbors is {1,2,3,4,5}. This notion is motivated by a problem in robotics. Fujita, Yamashita and Kameda showed
that every 3-regular graph has a (5,2)-configuration. In this talk, we will prove that except for four graphs, every
graph of minimum degree at least two which does not contain K_{1,6} as
an induced subgraph has a (5,2)-configuration. This is joint work with Waseem Abbas, Magnus Egerstedt, Robin Thomas, and Peter Whalen.

Series: Graph Theory Seminar

We will discuss how linear isoperimetric bounds in graph coloring lead to
new and interesting results. To that end, we say a family of graphs
embedded in surfaces is hyperbolic if for every graph in the family the
number of vertices inside an open disk is linear in the number of vertices
on the boundary of that disk. Similarly we say that a family is strongly
hyperbolic if the same holds for every annulus.
The concept of hyperbolicity unifies and simplifies a number of known
results about coloring graphs on surfaces while resolving some open
conjectures. For instance: we have shown that the number of
6-list-critical graphs embeddable on a fixed surface is finite, resolving a
conjecture of Thomassen from 1997; that there exists a linear time
algorithm for deciding 5-choosability on a fixed surface; that locally
planar graphs with distant precolored vertices are 5-choosable (which was
conjectured for planar graphs by Albertson in 1999 and recently resolved by
Dvorak, Lidicky, Mohar and Postle); that for every fixed surface, the
number of 5-list-colorings of a 5-choosable graph is exponential in the
number of vertices.
We may also adapt the theory to 3-coloring graphs of girth at least five on
surface to show that: the number of 4-list-critical graphs of girth at
least five on a fixed surface is finite; there exists a linear time
algorithm for deciding 3-choosability of graph of girth at least five on a
fixed surface; locally planar graphs of girth at least five whose cycles of
size four are far apart are 3-choosable (proved for the plane by Dvorak and
related to the recently settled Havel's conjecture for triangle-free graphs in the plane).
This is joint work with Robin Thomas.

Series: Graph Theory Seminar

Fifty years ago Erdos asked to determine the minimum number of $k$-cliques in a graph on $n$ vertices with independence number less than $l$ (we will refer this as $(k,l)$-problem). He conjectured that this minimum is
achieved by the disjoint union of $l-1$ complete graphs of size $\frac{n}{l-1}$.
This conjecture was disproved by Nikiforov who showed that Erdos' conjecture can be true only for finite many pairs of $(k,l)$. For $(4,3)$-problem, Nikiforov further conjectured that the balanced blow-up of a $5$-cycle achieves the minimum number of $4$-cliques.
We first sharpen Nikiforov's result and show that Erdos' conjecture is false whenever $k\ge 4$ or $k=3, l\ge 2074$. After introducing tools (including Flag Algebra) used in our proofs, we state our main theorems, which characterize the precise structure of extremal examples for $(3,4)$-problem and $(4,3)$-problem, confirming Erdos' conjecture for $(k,l)=(3,4)$ and Nikiforov's conjecture for $(k,l)=(4,3)$. We then focus on $(4,3)$-problem and sketch the proof how we use stability arguments to get the extremal graphs, the balanced blow-ups of $5$-cycle.
Joint work with Shagnik Das, Hao Huang, Humberto Naves and Benny Sudakov.