Final project description
Here below you find an outline of what this
file contains regarding the final project for
this course. By clicking on the each term, you
go directly to that section of the file.
As mention in the Syllabus, each of you is to write
a final report that is due on the final exam day. This
report will be read and graded by me.
The report should be about 3--8 pages, and absolutely no longer
than 10 pages! The report should be about a specific
topic in graph theory, combinatorics or discrete mathematics
in general. It is your choice what topic you pick, but
it is preferable to choose a topic satisfying the
- A topic in graph theory, which we have not covered, or
at least not covered in great detail, upon the end of
lecturing this coming December.
- A topic from a book or an article, which has not
been written by the instructor (Geir Agnarsson, in this case! :)
- A topic of technical flavor, rather than historical. (Although
history is some times interesting, when explaining certain technical
issues, i.e. for planar graphs and surface embeddings of graphs in
- A topic which would enhance your understanding of
graph theory or combinatorics. That is, you should have
the feeling of having learned something after
having written your report.
- A topic that you will find possible to explain to
a nonspecialist in fair detail. That is, you should feel
comfortable being able to answer any silly question
about your topic.
Below are themes and questions, that could serve as topics for your
project (these are only suggestions!):
- Some minor theorems:
Some minor theorems, by Niel Robertson and Paul Seymour,
have shed a new light on how one can possible go about
proving that certain computational problems can be
done in polynomial time, that is, the running time
is proportional to f(n), where f
is a polynomial and n is the number of
vertices of the graph in question. Discuss how some
of these theorems can be used, to determine the
existence of such polynomial time algorithms,
without actually constructing them.
Key words and phrases: Contraction, minor of a graph,
Robertson, Seymour, hereditary property of graphs.
- Click here for some highlights of the minor theorems by Robertson and Seymour,
written by one of the current heavy-weight champions in graph theory,
Robin Thomas at Georgia Tech.
- Click here for an article which describes
some algorithmic implications of a minor theorem.
- Algorithmic aspects of trees:
How do certain searches, like binary search, depth first
search and breath first search, all work? (a brief description.)
When is which best? How fast do they work? That is,
discuss (the best you can!) their computational
Key words and phrases: Tree, binary search tree,
decision, tree traversal, greedy algorithm,
running time, complexity.
- Click here for searching in a tree-like data structures.
- Click here for a summary of various greedy algorithms on trees.
- Hamiltonian cycles in graphs:
When we know that a certain graph G
has a Hamiltonian cycle (for example by Ore's theorem,
or by some other theorem ensuring a Hamiltonian cycle)
how fast is one able to determine an actual Hamiltonian
cycle? Is there a general lower bound, in terms of
the number of vertices n, and what is
the best running time you can find?
Key words and phrases: Hamiltonian graphs, spanning
cycle, deterministic algorithm, nondeterministic algorithm.
- Click here for a good survey paper on Hamiltonian graphs.
- Click here for many nice figures, done by Stephen Wolfram, the
creator of the software package MATHEMATICA, used by many universities.
- Traveling salesman problem:
Explain exactly what the traveling salesman problem (TSP)
is, discuss its history, why it is such computationally hard
problem and why researchers are still working on it (if it
has been proved that even its "yes/no" version
is NP-complete.) How good of an approximation can one
expect to obtain, if the running time is to be at most
Key words and phrases: traveling salesman problem,
deterministic algorithm, nondeterministic algorithm,
approximation algorithm, running time, complexity.
- Click here
for the "Home Page" of the traveling salesman problem (TSP).
- Click here for a list of papers and research articles on the TSP.
- The four color theorem:
Discuss the history if the four color theorem, what went
wrong in Kempe's proof and the approach of the proof
of Appeal, Haken and Koch. Then discuss the improved
approach of Robertson, Sanders, Seymour and Thomas.
What was improved? Are we any closer to a self-contained
Key words and phrases: Four color theorem,
four color conjecture, five color theorem, Kempe,
Appeal, Haken, Koch, Robertson, Sanders, Seymour, Thomas.
- Click here for a well written account of the improvement of the FCT by Robertson and co (now you know where I got the US map from! :)
- Click here for the actual article itself by Robertson and co.
- Various planarity parameters:
The genus is one parameter that measures how far a graph
is from being planar. There are other parameters that
measures that also. Do all these parameters follow each other?
That is, if, say, the genus of a given graph is large, are
then all the other planarity parameters also large?
Discuss which follow each other which not, how fast the grow, etc.
Key words and phrases: planar graph, torus, toroidal graph,
genus, crossing number, parameters of planarity, topological graph theory.
- Click here for a nice compilations of various problems regarding genera
(or "genuses") of graphs, crossing numbers and other problems in topological
- Click here for a discussion of the genus of a graph by Stephen Wolfram.
- Telecommunication systems and graphs:
As the number of people going "wireless", in the sense that
more and more people choose not to have a home phone, but
rather rely in their cellular phones, discuss how this wireless
network can be describe by a graph, directed graph, or some
"graph-like" structure that captures the essence of this model.
What difficulties arise when the number of people grows? Or
when cell phones are clustered tightly in large cities?
Key words and phrases: network, random graphs, internet
graphs, massive graphs, the power law.
- Click here for a
a nice paper on massive graphs, written by one of the for-runner
of such research today, Fan Chung.
- Click here for a
the wikipedia site with many key words and pointers.
- Partially ordered sets and graphs.
How are partially ordered sets (POSets) and graphs related, if at all?
Is there some property that a finite POSet can have,
such that its resulting graph has some properties that we
Key words and phrases: posets, acyclic graphs,
- Click here
for an interesting talk on interplay between posets, graphs and complexes.
- Click here
for a site on the Wolfram-Research on posets.
- Ramsey theory for graphs:
Describe what Ramsay theory for graphs is all about. Discuss
why so few concrete results have been established. Discuss
the corresponding Ramsey numbers.
Key words and phrases: Pigeon hole principle,
partition of a set, Ramsey theory, Ramsey numbers.
- Click here for a site on the Wolfram-Research on Ramsey theory (definitions and
- Click here
for the wiki site on the basics of Ramsey theory, the history of
Frank P. Ramsey and much more.
- Click here
for a good description of what Ramsey's theorem states in terms of graphs.
Some general literature:
Below is a list of some books that include
general graph theory, which either cover more than
we have covered in our class,
or include some special topics. Please note that
the description with each text is my (Geir Agnarsson)
- The discrete math book by Kenneth H. Rosen
is a basic, but a very comprehensive introductory text with
numerous exercises, some of which are nontrivial or
even open problems! (They are, of course, listed as such...)
Although it is an undergraduate text, this book
is an excellent text for self-study and quick look-up for
various definitions on discrete mathematics and basic graph
In addition, it also includes a gentle introduction to the
theory of computations and how directed graphs are used
to model various Finite-State-Machines and Turing-Machines.
- The combinatorics book by
van Lint and Wilson is a graduate text consisting of selected
topics in combinatorics and graph theory. It is a comprehensive
introduction to combinatorial mathematics and an excellent source
of examples and nontrivial, though solvable, problems.
- The graph theory book by Ron Gould is a well
written text, by an author who has equally much background in
computer science as in mathematics. This upper division undergraduate
text/ beginning graduate text is mathematically rigorous and include
numerous graph algorithms.
- The introductory book by Wilson on graph theory
is an excellent invitation to graph theory! This upper division undergraduate
text/beginning graduate text is short, concise and
written in the "get-right-to-the-point" style which
is common in scholarly texts in Europe.
The topics are well chosen and the book has a good variety of exercises.
- The graph theory book by Deo is one of the
first comprehensive texts on graph theory and its application
to various other disciplines, including computer science,
engineering (especially network analysis in electrical engineering)
and operations research. This beginning graduate text
emphasizes practical computational applications and includes
an early account of the use of graphs in the mentioned
fields. This is a classic text, which has been used as a classroom
text throughout the US and in India for decades.
- The graph theory book by Merris is a well
written and relatively short book on general graph theory.
This upper division undergraduate/beginning graduate is not
overly comprehensive, but is rather on excellent chosen
topics, and has a good source of exercises. The writing
is friendly and inviting despite the fact that it is
mathematically rigorous and covers some nontrivial topics
which are not too common in the graph theory literature.
This book includes an excellent discussion on chordal graphs,
interval graphs and a special class of interval graphs called
- The graph theory book by Diestel, is a beautiful
textbook, where graph theory is viewed as a branch of pure mathematics.
This graduate text emphasizes style, short but tricky proofs, existence
theorems rather than algorithmic ones. It is highly rigorous mathematically
and recommended for those who appreciate the beauty of graph theory
rather than its applications.
- The graph theory text by Bollobas, is a classic
well written text, which treats the topics it covers very well. This
graduate text emphasizes extremal graph theory and how
to obtain various existence theorems for graphs. The text is mathematically
rigorous and the topics as well as the exercises are very well chosen.
- The graph theory book
by Doug West is one of the most,
if not the most, comprehensive text on general graph theory in the literature
today. This book is bound to become a classic and is already a standard
reference in other books/articles on graph theory. This graduate level text
includes an excellent source of exercises and references. The text covers
pretty much every topic of current interest in graph theory. It is
mathematically rigorous and the title "Introduction to graph theory"
should rather be just "Graph theory".
- The graph theory book by Frank Harary is the
classic book on graph theory in general. Written by one of the current
heavy-weight champions in graph theory, it is mathematically rigorous
and includes all the classic topics of graph theory, except the most
Some books on special topics:
Below is a list of some books on some
special topics in graph theory or that strongly
relate to graph theory. Again, the opinion
written is mine (Geir Agnarsson) and not the
general wide-spread on (at least not to my knowledge.)
- The Art of computer programming treatise by Donald
(currently!) of three comprehensive volumes. These three books are probably the most
referenced books in algorithms and computer science in general and are the professionals
handbook. They are very well
written, and are the most comprehensive and mathematically rigorous texts
on algorithms that currently exits. Considered by many "the bible(s?) of
fundamental algorithms". There is a good section on trees
and tree-like data structures in Volume one.
The plan of the author is to write three more volumes (Volumes 4,5 and 6)
if time permits (it has taken many decades for him to write the first three!)
For additional information, you might want to check Donald's
for various questions that you might have about these books.
- The book
by Herbert Wilf is an excellent book on the use of generating functions
in combinatorics and enumerations in graph theory. You can download the
whole book (courtesy of Herbert and Academic Press!) from the mentioned
- the book by Riordan is a classic text on the use
of generating functions in numerous counting problems, including
enumerations of trees and various graphs. Despite its age (published
in 1958) the book still serves as a frequent reference in the graph
- The Four-Color Theorem by Rudolf and Gerda Fritsch,
is a well written and thorough account of the history of the Four-Color
Theorem/Conjecture. It emphasizes the topological difficulty regarding
planar graphs, something which is mostly ignored be many in combinatorics
and discrete mathematics. Makes an interesting and enriching reading.
It is a good idea to skip the proof toward the end of the text.
- The book
by Trotter on partially ordered sets, is
a comprehensive treatment, with an emphasis on order dimension of a poset.
Although published in 1992, many open problems listed and discussed there
are still unsolved. Contains a good list of references of related research
- The text book by Bernd Schroeder on partially ordered
sets in general is a well written book, with a broader coverage than
the above mentioned book by Trotter. This book covers more
recent topics regarding posets and the underlying graphs of posets.
It contains an excellent account of how posets related to topology
and other branches of mathematics.
- The book Matching Theory
by Lovasz and Plummer
has an excellent collection of algorithms and theory on matchings,
especially on matchings in bipartite graphs. It is, by many, considered
the "bible of matchings and matching algorithms". This book is written
by know heavy-weighters in graph theory. The book is out of print
and is hard to get by, but there is a
the George Washington University library.
Elements of writing-style:
Rule number one two and three:
Do not try to be more comprehensive than you can handle!
-- Write in as simple terms as possible,
and attempt to make the reading as interesting
and clear for the reader. Avoid displaying large formulas
if you can say what you want in plain English!