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
minimum degree (G). Describe a method of constructing
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). For a graph G with nonadjacent vertices u and v such
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
frequently assumed we were looking at a minimal example of
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 the adjacencies v1 v2 v3 v4 vk v1 . Let
e be the edge c
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, then there is an
edge e in G such that G e is still conn
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 , a2 , . . . , an , b1 , b2 , . . . , bn
and containing t
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 the indegree and outdegree respectively.
Note that d (v
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 facts about a directed graph D.
(a) (5 points) Prove that
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 below.
a2
b2
a1
b1
b0 a0
b3
a3
b4
a4
(a) (5 points) Demons
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 below.
(a) (5 points) Demonstrate that the Petersen graph
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 imitation.
(a) (5 points) Show that Hn,k is k -connected for e
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 Harary graphs are k -connected. You may use the
case presen
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
dwildstr@erdos.math.louisville.edu
(502)852-5845 (x5845)
Natural Sciences Building
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 simple but distinctly nontrivial example, to set the stag
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 graph you can nd which either avoids or satises some graph
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, so it is probably not surprising that one can investi
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) (4 points) It is known that the connectivity (G) of a g
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 approach is to prove the contrapositive via Kuratowskis
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
destroys all cycles in G. Prove that G is planar.
2. (10
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 useful class of graphs with several interesting propertie
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 if the vertex-set of G can be partitioned into two sets
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 vertices as possible into well-matched pairs. It is to this e
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 condition is in fact sucient.
Theorem 1 (Tutte, 47). If th
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 we shall see
that it is in fact a tree.
Proposition 1.
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 a graph. If we visualize a graph as a system of
pipelin
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 constructing a
maximal ow. Note that there is only one place in t
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 structure: if there are n competitors in a
tournament
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 known as Eulers Formula: if a planar projection of a conn
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 cfw_a1 , a2 , . . . , an , b1 , b2 , . . . , bn
and co