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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.

Series: Graph Theory Seminar

Associated to every finite graph G there is a canonical ideal which encodes the linear equivalences of divisors on G. In the study of this ideal the concept of "connected flags" arise naturally. The focus of this talk will be the study of combinatorial properties of these connected flags. This is a joint work with Fatemeh Mohammadi.
(This talk is related to the talk I gave on October 12th in the Combinatorics seminar, but I will not assume anything from the previous talk.)

Series: Graph Theory Seminar

A signed graph is a pair $(G,\Sigma)$ where $G$ is an undirected graph (in which
parallel edges are permitted, but loops are not) and $\Sigma \subseteq E(G)$.
The edges in $\Sigma$ are called odd and the other edges are called even. A
cycle of $G$ is called odd if it has an odd number of odd edges. If $U\subseteq
V(G)$, then re-signing $(G,\Sigma)$ on $U$ gives the signed graph
$(G,\Sigma\Delta \delta(U))$. A signed graph is a minor of $(G,\Sigma)$ if it
comes from $(G,\Sigma)$ by a series of re-signing, deletions of edges and
isolated vertices, and contractions of even edges.
If $(G,\Sigma)$ is a signed graph with $n$ vertices, $S(G,\Sigma)$ is the set of
all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j} > 0$ if $i$ and
$j$ are connected by only odd edges, $a_{i,j} < 0$ if $i$ and $j$ are connected
by only even edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by
both even and odd edges, $a_{i,j}=0$ if $i$ and $j$ are not connected by any
edges, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$.
The stable inertia set, $I_s(G,\Sigma)$, of a signed graph $(G,\Sigma)$ is the
set of all pairs $(p,q)$ such that there exists a matrix $A\in S(G,\Sigma)$
that has the Strong Arnold Hypothesis, and $p$ positive and $q$ negative
eigenvalues. The stable inertia set of a signed graph forms a generalization of
$\mu(G)$, $\nu(G)$ (introduced by Colin de Verdi\`ere), and $\xi(G)$
(introduced by Barioli, Fallat, and Hogben).
A specialization of $I_s(G,\Sigma)$ is $\nu(G,\Sigma)$, which is defined as the
maximum of the nullities of positive definite matrices $A\in S(G,\Sigma)$ that
have the Strong Arnold Hypothesis.
This invariant is closed under taking minors, and characterizes signed graphs
with no odd cycles as those signed graphs $(G,\Sigma)$ with $\nu(G,\Sigma)\leq
1$, and signed graphs with no odd-$K_4$- and no odd-$K^2_3$-minor as those
signed graphs $(G,\Sigma)$ with $\nu(G,\Sigma)\leq 2$. In this talk we will
discuss $I_s(G,\Sigma)$, $\nu(G,\Sigma)$ and these characterizations.
Joint work with Marina Arav, Frank Hall, and Zhongshan Li.

Series: Graph Theory Seminar

Over the past 40 years, researchers
have made many connections between the
dimension of posets and the issue of planarity
for graphs and diagrams, but there appears
to be little work connecting
dimension to structural graph theory. This situation
has changed dramatically in the last several months. At the Robin Thomas birthday conference, Gwenael
Joret, made the following striking conjecture, which
has now been turned into a theorem: The dimension
of a poset is bounded in terms of its height and the
tree-width of its cover graph. In this talk, I will present the proof of this result. The general contours of
the argument should be accessible to graph theorists and combinatorists (faculty and students) without deep knowledge of either dimension or tree-width.
The proof of the theorem was
accomplished by a team of six researchers: Gwenael Joret, Piotr Micek, Kevin Milans, Tom Trotter,
Bartosz Walczak and Ruidong Wang.

Series: Graph Theory Seminar

We show that any n-vertex complete graph with edges colored with three
colors contains a set of at most four vertices such that the number of the
neighbors of these vertices in one of the colors is at least 2n/3. The
previous best value proved by Erdos et al in 1989 is 22. It is conjectured
that three vertices suffice. This is joint work with Daniel Kral,
Chun-Hung Liu, Jean-Sebastien Sereni, and Zelealem Yilma.

Series: Graph Theory Seminar

A graph $G$ contains a graph $H$ as an immersion if there exist distinct vertices $\pi(v) \in V(G)$ for every vertex $v \in V(H)$ and paths $P(e)$ in $G$ for every $e \in E(H)$ such that the path $P(uv)$ connects the vertices $\pi(u)$ and $\pi(v)$ in $G$ and furthermore the paths $\{P(e):e \in E(H)\}$ are pairwise edge disjoint. Thus, graph immersion can be thought of as a generalization of subdivision containment where the paths linking the pairs of branch vertices are required to be pairwise edge disjoint instead of pairwise internally vertex disjoint. We will present a simple structure theorem for graphs excluding a fixed $K_t$ as an immersion. The structure theorem gives rise to a model of tree-decompositions based on edge cuts instead of vertex cuts. We call these decompositions tree-cut decompositions, and give an appropriate definition for the width of such a decomposition. We will present a ``grid" theorem for graph immersions with respect to the tree-cut width.
This is joint work with Paul Seymour.

Series: Graph Theory Seminar

In the node-connectivity augmentation problem, we want to add a minimum
number of new edges to an undirected graph to make it k-node-connected.
The complexity of this question is still open, although the analogous questions
of both directed and undirected edge-connectivity and directed
node-connectivity augmentation are known to be polynomially solvable.
I present a min-max formula and a polynomial time algorithm for the
special case when the input graph is already (k-1)-connected. The formula has
been conjectured by Frank and Jordan in 1994.
In the first lecture, I presented previous results on the other connectivity augmentation variants.
In the second part, I shall present my min-max formula and the main ideas of the proof.

Series: Graph Theory Seminar

In the node-connectivity augmentation problem, we want to add a minimum
number of new edges to an undirected graph to make it k-node-connected.
The complexity of this question is still open, although the analogous questions
of both directed and undirected edge-connectivity and directed
node-connectivity augmentation are known to be polynomially solvable.
I present a min-max formula and a polynomial time algorithm for the
special case when the input graph is already (k-1)-connected. The formula has
been conjectured by Frank and Jordan in 1994.
In the first lecture, I shall investigate the background, present some results on the previously
solved connectivity augmentation cases, and exhibit examples motivating the complicated
min-max formula of my paper.

Series: Graph Theory Seminar