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Unformatted text preview: A little-bit of Graph Theory Periklis A. Papakonstantinou ? University of Toronto We consider the problem of extracting global properties for a graph by just knowing some local information about it. This will take us to the first simpler but yet remarkable developments in graph theory. In particular, when knowing a lower bound on the degree of vertices of a simple graph we show that such a graph always has a “large” circuit (function of the degree of the vertices). Graph theory is one of the most exciting areas of combinatorics. It is true that people can very quickly get the definitions and get hands on work almost immediately. It is also true that for the basic developments almost no previous knowledge on discrete math is needed. This is justified by the fact that a first discrete math course can (and sometimes does) only deal with basic graph theory. The definition of an undirected graph is given in your text and the first recitation. Also Epp’s text gives the definition of a circuit in a graph. An undirected graph is called simple if there are no multiple edges between two vertices or self-loops (edges from a vertex v to v ). In this document we restrict our attention to simple undirected graphs which are of the great interest with respect to applications both in mathematics and engineering. There are computational graph problems that are known efficient algorithms to solve them. For example, in order to test whether in a graph on n vertices there is a circuit of length 1 3 (also known as triangle) one can simply check every set of three vertices. There are only ( n 3 ) ≈ n 3 such sets which means that such an algorithm works in polynomial time in the number the graph vertices (i.e. it’s an efficient algorithm). Similar things hold when we want to check the existence of circuits of any constant length. But how about when we wish to check for circuits of non-constant length? For example, what if we want to check for Hamiltonian circuits or for circuits whose length grows with n where n is the number of vertices of the graph (here clearly we consider a family of graphs one for each n )? Is there still an efficient algorithm? A widely believed conjecture that comes under the title “ P vs NP ” is associated with these questions. We will not try to answer this question here but if you want to see what it is about and also a nice way to see how one can make $ 1,000,000 (US) you may check this http://www.claymath.org/millennium/....
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This note was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto.
- Fall '06