Graph Theory Reading Group

1
Graph Theory Reading Group
Meeting 1 Overview and Agenda
2
Topics in Graph Theory
Definitions and basic properties Planarity and
embeddings Flows and Matchings Extremal
Problems Coloring Random Graphs Algebraic Graph
Theory (Algorithms, Intractability)
Books BB Bela Bolobas, Modern Graph Theory. AG
Alan Gibbons, Algorithmic Graph Theory.
3
Definitions and Basic Properties

• types of graphs
• subgraphs
• isomorphisms
• paths, circuits, cycles
• trees, spanning trees
• bipartite graphs
• Hamiltonian and Euler cycles
• Kirchoffs laws?
• Software?
• etc.

4
Planarity and Embeddings
K4 is planar
K5 is not
Eulers formula Kuratowskis theorem Planarity
algorithms
Other questions when is one embedding deformable
to another? Chirality.
5
Flows and Matchings
3
6
7
t
5
2
1
4
1
s
5
3
9
girls
boys

• Menegers theorem (separating vertices)
• Halls theorem (when is there a matching?)
• Stable matchings
• Various extensions and similar problems
• Algorithms

6
Extremal Problems
e.g. Given a graph G, how many edges in a graph
of size n guarantee that it contains G? e.g.
Given G(n,m), at most what is the girth and at
least what is the circumference?
Erdos-Stone Fundamental Theorem of Extremal
Graph Theory ?n2 edges guarantee Kr1 and
Kr1(t).
BB IV could take several weeks.
7
Coloring

• How do you assign a color to each vertex so that
  adjacent vertices are colored differently?
• Chromatic number of certain types of graphs.
• k-Coloring is NP Complete.
• Edge coloring

BB V AG Ch. 7
8
Random Graphs

• Form probability spaces containing graphs or
  sequences of graphs as points.
• Simple properties of almost all graphs.
• Phase transition as you add edges component size
  jumps from log(n) to cn.

BB VII
9
Algebraic Graph Theory
a
a3
a2
group elements
a
a

• Cayley diagrams
• Adjacency and Laplacian Matrices their
  eigenvalues and the structure of various classes
  of graphs

a
1
a
generators
BB VIII --- there are 4 sections, could take a
while. There are entire books on the subject that
might be better. Alex would know.
10
Algorithms

• DFS, BFS, Dijkstras Algorithm...
• Maximal Spanning Tree...
• Planarity testing, drawing...
• Max flow...
• Finding matchings...
• Finding paths and circuits...
• Traveling salesperson algorithms...
• Coloring algorithms...

11
Other Topics?
Graph partitioning(look up) Transitive
closures Topological Graphs(Elon) Matroids and
greedy algorithms (jason has refs)
12
Agenda and Assignments
Receding horizon lets always know who will lead
the next five meetings.
1. BB I.1, I.2, I.3 (Basics) Dave 2. BB I.4
AG 3 (Planarity) Jason 3. AG 4 (Flows) Bill 4.
AG 5 (Matchings) Cristian 5. BB II.3 and VIII.2
(Algebra) Lars
BB III
Who else Steven Low Ask Burdick, Bruck, Effros,
Hassibi if they have people who want to join.