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.

Brief Info: 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 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.

Suggestions: Below are themes and questions, that could serve as topics for your project (these are only suggestions!):

  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:

  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:

  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:

  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:

  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:

  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:

  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:

  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:

  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:

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) personal opinion.

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.)

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!

Best wishes!

Geir Agnarsson