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Series: Graph Theory Seminar

We use K_4^- to denote the graph obtained from K_4 by removing an edge,and
use TK_5 to denote a subdivision of K_5. Let G be a 5-connected nonplanar
graph and {x_1, x_2, y_1, y_2} \subseteq V (G) such that G[{x_1,x_2, y_1,
y_2}] = K_4^- with y_1y_2 \in E(G). Let w_1,w_2,w_3 \in N(y_2)- {x_1,x_2}
be distinct. We show that G contains a TK_5 in which y_2 is not a branch
vertex, or G - y_2 contains K_4^-, or G has a special 5-separation, or G'
:= G - {y_2v : v \in {w_1,w_2,w_3, x_1, x_2}} contains TK_5.In this talk,
we will obtain a substructure in G' and several additional paths in G', and
then use this substructure to find the desired TK_5.

Series: Graph Theory Seminar

We use K_4^- to denote the graph obtained from K_4 by removing an edge,and use TK_5 to denote a subdivision of K_5. Let G be a 5-connected nonplanar graph and {x_1, x_2, y_1, y_2} \subseteq V (G) such that G[{x_1,x_2, y_1, y_2}] = K_4^- with y_1y_2 \in E(G). Let w_1,w_2,w_3 \in N(y_2)- {x_1,x_2} be distinct. We show that G contains a TK_5 in which y_2 is not a branch vertex, or G - y_2 contains K_4^-, or G has a special 5-separation, or G - {y_2v : v \in {w_1,w_2,w_3, x_1, x_2}} contains TK_5.In this talk, we will show the existence of a path X in G whose removal does not affect connectivity too much.

Series: Graph Theory Seminar

Seymour and, independently, Kelmans conjectured in the 1970s that
every 5-connected nonplanar graph contains a subdivision of K_5. This
conjecture was proved by Ma and Yu for graphs containing K_4^-. In order to
establish the Kelmans-Seymour conjecture for all graphs, we need to
consider 5-separations and 6-separations with less restrictive structures.
We will talk about special 5-separations and 6-separations whose cut
contains a triangle. Results will be used in subsequently to prove the
Kelmans-Seymour conjecture.

Series: Graph Theory Seminar

Seymour and, independently, Kelmans conjectured in the 1970s that
every 5-connected nonplanar graph contains a subdivision of K_5. This
conjecture was proved by Ma and Yu for graphs containing K_4^-, and an
important step in their proof is to deal with a 5-separation in the graph
with a planar side. In order to establish the Kelmans-Seymour conjecture
for all graphs, we need to consider 5-separations and 6-separations with
less restrictive structures. We will talk about special 5-separations and
6-separations, including those with an apex side. Results will be used in
subsequently to prove the Kelmans-Seymour conjecture.

Series: Graph Theory Seminar

The goal of this talk is to show recent advances regarding two important
mathematical problems. The first one can be straightforwardly formulated in
a graph theory language, but can be possibly applied in other fields. The
second one was motivated by machine learning applications, but leads to
graph theory techniques.
The celebrated open conjecture of Erdos and Hajnal from 1989 states
that families of graphs not having some given graph H as an induced
subgraph contain polynomial-size cliques/stable sets (in the undirected
setting) or transitive subsets (in the directed setting). Recent techniques
developed over last few years provided the proof of the conjecture for new
infinite classes of graphs (in particular the first infinite class of prime
graphs). Furthermore, they gave tight asymptotics for the Erdos-Hajnal
coefficients for many classes of prime tournaments as well as the proof of
the conjecture for all but one tournament on at most six vertices and the
proof of the weaker version of the conjecture for trees on at most six
vertices. In this part of the talk I will summarize these recent
achievements.
Structured non-linear graph-based hashing is motivated by applications in
neural networks, where matrices of linear projections are constrained to
have a specific structured form. This drastically reduces the size of the
model and speeds up computations. I will show how the properties of the
underlying graph encoding correlations between entries of these matrices
(such as its chromatic number) imply the quality of the entire non-linear
hashing mechanism. Furthermore, I will explain how general structured
matrices that very recently attracted researchers’ attention naturally lead
to the underlying graph theory description.

Series: Graph Theory Seminar

There has been extensive research on cycle lengths in graphs with large
minimum degree. In this talk, we will present several new and tight results
in this area. Let G be a graph with minimum degree at least k+1. We
prove that if G is bipartite, then there are k cycles in G whose
lengths form an arithmetic progression with common difference two. For
general graph G, we show that G contains \lfloor k/2\rfloor cycles
with consecutive even lengths, and in addition, if G is 2-connected and
non-bipartite, then G contains \lfloor k/2\rfloor cycles with
consecutive odd lengths. Thomassen (1983) made two conjectures on cycle
lengths modulo a fixed integer k: (1) every graph with minimum degree at
least k+1 contains cycles of all even lengths modulo k; (2) every
2-connected non-bipartite graph with minimum degree at least $k+1$ contains
cycles of all lengths modulo k. These two conjectures, if true, are best
possible. Our results confirm both conjectures!
when k is even. And when k is odd, we show that minimum degree at
least $+4 suffices. Moreover, our results derive new upper bounds of the
chromatic number in terms of the longest sequence of cycles with
consecutive (even or odd) lengths. This is a joint work with Chun-Hung Liu.

Series: Graph Theory Seminar

A set F of graphs has the Erdos-Posa property if there exists a function
f such that every graph either contains k disjoint subgraphs each
isomorphic to a member in F or contains at most f(k) vertices
intersecting all such subgraphs. In this talk I will address the
Erdos-Posa property with respect to three closely related graph
containment relations: minor, topological minor, and immersion. We
denote the set of graphs containing H as a minor, topological minor and
immersion by M(H),T(H) and I(H), respectively.
Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa
property if and only if H is planar. And they left the question for
characterizing H in which T(H) has the Erdos-Posa property in the same
paper. This characterization is expected to be complicated as T(H) has
no Erdos-Posa property even for some tree H. In this talk, I will
present joint work with Postle and Wollan for providing such a
characterization. For immersions, it is more reasonable to consider an
edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs
and covering them by edges. I(H) has no this edge-variant of the
Erdos-Posa property even for some tree H. However, I will prove that
I(H) has the edge-variant of the Erdos-Posa property for every graph H
if the host graphs are restricted to be 4-edge-connected. The
4-edge-connectivity cannot be replaced by the 3-edge-connectivity.

Series: Graph Theory Seminar

We discuss the relationship between the chromatic number (Chi),
the clique number (Omega) and maximum average degree (MAD).

Series: Graph Theory Seminar

For integers k>=1 and n>=2k+1, the bipartite Kneser graph H(n,k) is defined
as the graph that has as vertices all k-element and all (n-k)-element
subsets of {1,2,...,n}, with an edge between any two vertices (=sets) where
one is a subset of the other. It has long been conjectured that all
bipartite Kneser graphs have a Hamilton cycle. The special case of this
conjecture concerning the Hamiltonicity of the graph H(2k+1,k) became known
as the 'middle levels conjecture' or 'revolving door conjecture', and has
attracted particular attention over the last 30 years. One of the
motivations for tackling these problems is an even more general conjecture
due to Lovasz, which asserts that in fact every connected vertex-transitive
graph (as e.g. H(n,k)) has a Hamilton cycle (apart from five exceptional
graphs).
Last week I presented a (rather technical) proof of the middle levels
conjecture. In this talk I present a simple and short proof that all
bipartite Kneser graphs H(n,k) have a Hamilton cycle (assuming that
H(2k+1,k) has one). No prior knowledge will be assumed for this talk
(having attended the first talk is not a prerequisite).
This is joint work with Pascal Su (ETH Zurich).

Series: Graph Theory Seminar

In the combinatorics of posets, many theorems are in pairs, one for chains
and one for antichains. Typically, the statements are exactly the same when
roles are reversed, but the proofs are quite different. The classic pair of
theorems due to Dilworth and Mirsky were the starting point for this
pattern, followed by the more general pair known respectively as the
Greene-Kleitman and Greene theorems dealing with saturated partitions. More
recently, a new pair has been discovered dealing with matchings in the
comparability and incomparability graphs of a poset. We show that if the
dimension of a poset P is d and d is at least 3, then there is a matching
of size d in the comparability graph of P, and a matching of size d in the
incomparability graph of P.