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Series: Graph Theory Seminar

A special feature possessed by the graphs of social networks
is triangle-dense. R. Gupta, T. Roughgarden and C. Seshadhri give a
polynomial time graph algorithm to decompose a triangle-dense graph into
some clusters preserving high edge density and high triangle density in
each cluster with respect to the original graph and each cluster has
radius 2. And high proportion of triangles of the original graph are
preserved in these clusters. Furthermore, if high proportion of edges in
the original graph is "locally triangle-dense", then additionally, high proportion of
edges of the original graph are preserved in these clusters.
In this talk, I will present most part of the paper "Decomposition of Triangle-dense Graphs"
in SIAM J. COMPUT. Vol. 45, No. 2, pp. 197–215, 2016, by R. Gupta, T. Roughgarden and C. Seshadhri.

Series: Graph Theory Seminar

Let G be a 5-connected nonplanar graph. To show the Kelmans-Seymour
conjecture, we keep contracting a connected subgraph on a special vertex z
until the following happens: H does not contain K_4^-, and for any subgraph
T of H such that z is a vertex in T and T is K_2 or K_3, H/T is not
5-connected. In this talk, we study the structure of these 5-separations
and 6-separations, and prove the Kelmans-Seymour conjecture.

Series: Graph Theory Seminar

Let G be a 5-connected nonplanar graph. To show the Kelmans-Seymour
conjecture, we keep contracting a connected subgraph on a special vertex z
until the following happens: H does not contain K_4^-, and for any subgraph
T of H such that z is a vertex in T and T is K_2 or K_3, H/T is not
5-connected. In this talk, we prove a lemma using the characterization of
three paths with designated ends, which will be used in the proof of the
Kelmans-Seymour conjecture.

Series: Graph Theory Seminar

Let G be a 5-connected graph and let x1, x2,y1,y2 in V(G) be distinct, such that G[{x1, x2, y1, y2}] is isomorphic to K_4^- and y1y2 is not in E(G). We show that G contains a K_4^- in which x1 is of degree 2, or G-x1 contains K_4^-, or G contains a TK_5 in which x1 is not a branch vertex, or {x2, y1, y2} may be chosen so that for any distinct w1,w2 in N(x1) - {x2, y1, y2}, G - {x1v : v is not in {w1, w2, x2, y1,y2} } contains TK_5.

Series: Graph Theory Seminar

In 1998, Reed proved that the chromatic number of a graph is
bounded away from its trivial upper bound, its maximum degree plus one, and
towards its trivial lower bound, its clique number. Reed also conjectured
that the chromatic number is at most halfway in between these two bounds.
We prove that for large maximum degree, that the chromatic number is at
least 1/25th in between. Joint work with Marthe Bonamy and Tom Perrett.

Series: Graph Theory Seminar

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.