### Final project description

Outline: 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.

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 following:

1. 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.
2. A topic from a book or an article, which has not been written by the instructor (Geir Agnarsson, in this case! :)
3. 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 general.)
4. 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.
5. 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.

1. 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.

Some websites:

• 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.

2. 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 complexities.

Key words and phrases: Tree, binary search tree, decision, tree traversal, greedy algorithm, running time, complexity.

Some websites:

• Click here for a summary of various greedy algorithms on trees.

3. 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.

Some websites:

• Click here for many nice figures, done by Stephen Wolfram, the creator of the software package MATHEMATICA, used by many universities.

4. 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 O(nk)?

Key words and phrases: traveling salesman problem, deterministic algorithm, nondeterministic algorithm, approximation algorithm, running time, complexity.

Some websites:

• Click here for a list of papers and research articles on the TSP.

5. 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 readable proof?

Key words and phrases: Four color theorem, four color conjecture, five color theorem, Kempe, Appeal, Haken, Koch, Robertson, Sanders, Seymour, Thomas.

Some websites:

• 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.

6. 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.

Some websites:

• Click here for a nice compilations of various problems regarding genera (or "genuses") of graphs, crossing numbers and other problems in topological graph theory.
• Click here for a discussion of the genus of a graph by Stephen Wolfram.

7. 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.

Some websites:

• 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.

8. 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 can describe?

Key words and phrases: posets, acyclic graphs, planar posets.

Some websites:

• Click here for an interesting talk on interplay between posets, graphs and complexes.

9. 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.

Some websites:

• Click here for a site on the Wolfram-Research on Ramsey theory (definitions and Ramsey numbers.)
• 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.

• 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 theory.

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 threshold graphs.

• 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 recent ones.

• The Art of computer programming treatise by Donald Knuth, consists (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 home page 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 website.

• 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 theory literature.

• 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 results.

• 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 copy at the George Washington University library.