Posts Tagged ‘Graph Theory’

First Montreal Spring School in Graph Theory

May 13, 2010

I am attending this one-month summer school in Structural Graph Theory, which is held in Montreal. It is a high-level school with world-class speakers, and I am really enjoying it. We have two classes everyday for four weeks, making a total of 40 lectues.

Paul Seymour talks about excluding induced subgraphs, and, in particular, will sketch a proof of the Strong Perfect Graph Theorem, which he (together with Chudnovsky, Robertson and Thomas) has proved in 2001. This was a huge result, with a 150-pages proof. Not only is he a great problem-solver, but also he is a great teacher. I really do enjoy sitting in his class. He is so smart, draws lots of pictures, and always proves what he wants in a bottom-up way; That is, unlike many others who first write a Theorem statement formally and then write the proof, he first describes intuitively why such a Theorem has to be true, and only then writes it down.

Maria Chudnovsky (the picture is a bit old) also talks about excluding induced subgraphs, but her lectures are independent of the Paul’s. She is also smart and a good instructor (she was Paul’s PhD student and must have learnt a lot from him).

Bruce Reed talks about excluding minors from graphs. In particular he talks about Robertson and Seymour’s Graphs Minor Project. He is a great researcher. Sometimes his lectures are confusing for me, in particular when he talks about planar graphs. Fortunately he has now turned to tree decompositions, which I am a little familiar with.

To summarize, I am so glad that I am attending this school. Structural Graph Theory is really nice (though hard to prove new results) and I enjoy sitting in the classes and working on the assignments. It is like I am back in the IMO training camp (7 years ago)! The city of Montreal is also very nice. Here you see a photo I took in the botanical garden, which I went to see on May 8th.

my approximation algorithms talk and some other news

April 8, 2010

  1. Yesterday I had a talk for the “approximation algorithms” course. The title of my talk was “Approximating the Number of Perfect Matchings in Bipartite Graphs” and I presented an algorithm of Jerrum and Sinclair, developed in 1988. I had spent a lot of time (3-4 days) preparing the slides (actually for this talk I learned to use the Beamer package to create LaTeX-based presentations, and the TikZ/PGF package for creating the graphics) and the result was very good, I liked my slides, and I got positive feedback from the audience. The slides can be found here. In a part of the algorithm I described, a Markov chain was used, and to introduce the concept to the audience, I used the intuition of “random walk on a graph” instead of the formal definition. The picture you see above was one of the slides used to illustrate the distribution of a random walker (a happy ghost in this picture) on the vertices of the underlying graph. I was really relieved after the talk since I am not used to present in English.
  2. The paper I posted about two months ago was accepted in Ars Combinatoria. Another paper which I worked on in the same team (when I was an undergrad) was also recently accepted to appear in Graphs and Combinatorics after some revisions, yoohooo !
  3. The Winter 2010 term is almost finished – I have one final exam (integer programming) and I have to submit the final report for approximation algorithms. I spent most of my time on the courses, and I also read a few papers of Nick (my supervisor).
  4. My plan for the next term: I will attend the First Montreal Spring School in Graph Theory in May, and when I return, I will focus on my research (random graphs) and read related stuff (some ideas: The probabilistic methods by Alon-Spencer, Randomized Algorithms by Motwani-Raghavan, Lectures on Random Graphs and Survey on Regular Random graphs by Nick). I won’t take any course but may audit computational geometry from computer science. What a pity I didn’t attend the “Multi-agent Systems” course offered in Winter 2010 in CS.
  5. A graph theory problem: Prove that there exists an integer k, such that for any connected graph G with more than 2 vertices, one can assign a label from {1,2,…,k} to each edge, such that the sum of the labels of edges incident to every vertex defines a proper coloring for the graph. It has been conjectured that k=3 works, and the currently best known bound is k=5. I think it is a cute problem, see here for more info.

Revised “Nowhere-zero Unoriented Flows in Hamiltonian Graphs” submitted

February 2, 2010

This work was done almost two years ago, when I was an undergrad in Sharif. After being rejected two times, the Ars Combinatoria journal accepted to publish it after revisions. The referee have made quite a few comments/suggestions. I was modifying it during the last 12 days. Today the final (revised) version was created, approved by all authors and sent to the corresponding author. If accepted, this would be my first publication. Well done and congratulations!

“Graph Theory” as seen by Bondy and Murty

January 27, 2010

“Like number theory, graph theory is conceptually simple, yet gives rise to challenging unsolved problems. Like geometry, it is visually pleasing. These two aspects, along with its diverse applications, make graph theory an ideal subject for inclusion in mathematical curricula.”

J. A. Bondy and U. S. R. Murty, September 2007

(Preface of their book “Graph Theory”)

Thomas’ tutorial and CO754 Assignment2

January 20, 2010

I have recently studied Thomas’ lecture notes on tree-decompositions on graphs, which gives the reader an insight into the concepts of tree-decomposition, tree-width etc. It had a understand-through-solving-problems approach that I believe¬† is very nice and instructive:

http://people.math.gatech.edu/~thomas/PAP/treenotes.pdf

But the interesting part was that once I saw the problems of the 2nd assignment of approximation algorithms, I found that one of the optional problems is straightforward as the idea was covered in the notes. Specifically, problem 4(b) is easy if you study paragraphs C5-C9 of the notes.