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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (firstname.lastname@example.org) March 02, 2010 Lecture 12: The Probabilistic Method The probabilistic method is an elegant and powerful technique for analysis of algorithms and combinatorial structures. It is based on a simple premise: in order to prove the existence of a structure, we merely need to show that there is a positive probability that the structure exists. This method was first used by Szele in 1943, but Erd os was the first to fully realize the power of this method and apply it to a wide variety of problems. 1 Chromatic Number of a Graph A fundamental question in graph theory is the relationship between the subgraphs of a graph and its chro- matic number. For example, can we show that triangle free graphs must have a chromatic number bounded by some constant k ? Recall that a proper vertex k-coloring of a graph G ( V,E ) is an assignment of k colors to vertices such that no two vertices of the same color share an edge. Theno two vertices of the same color share an edge....
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- Winter '09