MATH 682
Exam #1
1. (6 of 12 students attempted this) Answer the following questions related to boundsubverting examples:
(a) It is known that the connectivity (G) of a graph G is bounded above by the
MATH 682
1
1.1
Notes
Combinatorics and Graph Theory II
Hamiltonian properties
Hamiltonian Cycles
Last time we saw this generalization of Diracs result, which we shall prove now.
Proposition 1 (Ore 60)
MATH 682
1
1.1
Notes
Combinatorics and Graph Theory II
Advanced Chromatic Properties
Color-Criticality
When we were proving particular results in chromatic number, such as the ve-color theorem, we
fre
MATH 682
Problem Set #1 Solutions
1. (10 points) Prove that if graph G is connected and contains a cycle, then there is an
edge e in G such that G e is still connected.
Let a cycle in G be denoted by
MATH 682
Problem Set #1
This problem set is due at the beginning of class on January 21. Below, graph means
simple nite graph.
1. (10 points) Prove that if graph G is connected and contains a cycle, t
MATH 682
Problem Set #2
This problem set is due at the beginning of class on February 4. Below, graph means
simple nite graph.
1. (10 points) Let Kn,n be the bipartite graph with vertex set cfw_a1 , a
MATH 682
Problem Set #3 Solutions
1. (15 points) Demonstrate the following facts about a directed graph D.
(a) (5 points) Prove that vV (D) d (v ) = vV (D) d+ (v ). Recall that d and d+
D
D
represent
MATH 682
Problem Set #3
This problem set is due at the beginning of class on February 25. Below, graph means
simple nite graph except where otherwise noted.
1. (15 points) Demonstrate the following fa
MATH 682
Problem Set #4
This problem set is due at the beginning of class on March 23. Below, graph means
simple nite graph except where otherwise noted.
1. (10 points) The Petersen graph is shown bel
MATH 682
Problem Set #4
This problem set is due at the beginning of class on March 23. Below, graph means
simple nite graph except where otherwise noted.
1. (10 points) The Petersen graph is shown bel
MATH 682
Problem Set #5 Solutions
1. (10 points) Complete the proof that the Harary graphs are k -connected. You may use the case
presented in class of even values of k , either by citation or imitati
MATH 682
Problem Set #5
This problem set is due at the beginning of class on April 6. Below, graph means simple nite
graph except where otherwise noted.
1. (10 points) Complete the proof that the Hara
MATH 682
Syllabus
Combinatorics and Graph Theory II
Course Information
Name:
E-mail address:
Phone number:
Instructor:
Oce:
Oce hours:
Alternative oce hours:
Jake Wildstrom
[email protected]
MATH 682
1
1.1
Notes
Combinatorics and Graph Theory II
Ramsey Theory
Classical Ramsey numbers
Furthermore, there is a beautiful recurrence to give bounds on Ramsey numbers, but we will start
with a si
MATH 682
1
Notes
Combinatorics and Graph Theory II
Minimal Examples and Extremal Problems
Minimal and extremal problems are really variations on the same question: what is the largest or
smallest grap
MATH 682
1
1.1
Notes
Combinatorics and Graph Theory II
Other coloring problems
Edge coloring
Many properties which are traditionally based on vertices (e.g. connectivity) also exist in an edge
version
MATH 682
Exam #1
Answer exactly four of the following six questions. Indicate which four you would like
graded!
1. (10 points) Answer the following questions related to bound-subverting examples:
(a)
MATH 682
Exam #2 Solutions
1. (8 students attempted this problem) Suppose that G is a simple graph that contains two edges whose removal destroys all cycles in G. Prove that G is planar.
The easiest a
MATH 682
Exam #2
Answer exactly four of the following six questions. Indicate which four you would like
graded!
1. (10 points) Suppose that G is a simple graph that contains two edges whose removal
de
MATH 682
1
Notes
Combinatorics and Graph Theory II
Trees
Recall that we nished the last semester with the introduction of connectedness and connected components. This leads us to an interesting and us
MATH 682
1
Notes
Combinatorics and Graph Theory II
Bipartite graphs
One interesting class of graphs rather akin to trees and acyclic graphs is the bipartite graph:
Denition 1. A graph G is bipartite i
MATH 682
1
Notes
Combinatorics and Graph Theory II
Matchings
A popular question to be asked on graphs, if graphs represent some sort of compatability or association, is how to associate as many vertic
MATH 682
1
Notes
Combinatorics and Graph Theory II
Matchings with Tuttes Theorem
Last week we saw a fairly strong necessary criterion for a graph to have a perfect matching. Today
we see that this con
MATH 682
1
Notes
Combinatorics and Graph Theory II
Local and Global 2-connectedness
1.1
Block structure, concluded
Previously we saw that the block structure of a connected graph was connected; today
MATH 682
1
Notes
Combinatorics and Graph Theory II
Flows
A logical followup (or predecessor) to Mengers theorem, and its associated discussion of multiple
simultaneous routes, is the concept of ow in
MATH 682
1
Notes
Combinatorics and Graph Theory II
Flows, continued
1.1
The Ford-Fulkerson algorithm
The proof above contains, as a subtle sidenote, the outline of an explicit algorithm for constructi
MATH 682
1
Notes
Combinatorics and Graph Theory II
Tournaments
Denition 1. An orientation of the complete undirected graph is called a tournament.
The denition above is of a natural sort of real-world
MATH 682
1
1.1
Notes
Combinatorics and Graph Theory II
Planar graphs
More fun with faces
We developed this idea of a face of a planar projection of a graph G, which motivated the very
useful result kn
MATH 682
Problem Set #2 Solutions
This problem set is due at the beginning of class on February 4. Below, graph means
simple nite graph.
1. (10 points) Let Kn,n be the bipartite graph with vertex set