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Lecture16

# Lecture16 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) 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. The

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