chapter-1[1] - APPLIED COMBINATORICS CHAPTER 1: AN OVERVIEW...

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Unformatted text preview: APPLIED COMBINATORICS CHAPTER 1: AN OVERVIEW WILLIAM T. TROTTER AND MITCH KELLER Abstract. Welcome to Math 3012 - Applied Combinatorics!! As we hope you will sense right from day one in this class, we believe combinatorial mathematics is one of the most fascinating and cap- tivating subjects on the planet. Combinatorics is very concrete and has a wide range of applications, but it also has an intellectu- ally appealing theoretical side. In our course, you will get a taste of both. Computing will play an important role in the course, as we want you to learn how algorithms and recursive processes are used to get things done. We will do some programming in C and we will use the Maple software package. Dont worry if you are not familiar with programming languages or mathematical software. All the necessary tools will be developed from scratch. Please note that we will not be using a textbook in this section, so if you have purchased one, you might want to consider taking it back to the bookstore. Instead, we will be using lecture notes posted on the course web-site and distributed by email, like this document. 1. Introduction There are three prinicpal themes to our course: (1) Discrete Structures: Graphs, digraphs, networks, designs, posets, strings, patterns, distributions, coverings, partitions. (2) Enumeration: permutations, combinations, inclusion/exclusion; generating functions; recurrence relations, Polya counting. (3) Algorithms and Optimization: Sorting, spanning trees, short- est paths, eulerian circuits, hamitonian cycles, graph coloring, planarity testing, network flows, bipartite matchings, chain par- titions. To illustrate the accessible, concrete nature of combinatorics and to motivate topics that we will study during the course, we are includ- ing in this preliminary chapter a first look at combinatorial problems, choosing examples from graph theory, number theory and optimization. Date : January 1, 2006. 1 2 TROTTER AND KELLER The discussion is very informalbut this should serve to explain why we have to be more precise at later stages. We ask lots of questions, but at this stage, youll only be able to answer a few. Later, youll be able to answer many more ...but as promised earlier, most likely youll never be able to answer them all. And if were wrong in making that statement, then youre certain to become very, very famous. Our discussion will also introduce you to Alice and Bob, who are almost always on opposite sides of any issue. So lets begin. 2. Combinatorics and Graph Theory A graph G consists of a vertex set V and a collection E of 2-element subsets of V . Elements of E are called edges. In our course, we will (almost always) use the convention that V = { 1 , 2 , 3 ,...,n } for some positive integer n . With this convention, graphs can be described pre- cisely with a text file: (1) The first line of the file contains a single integer n , the number of vertices in the graph....
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chapter-1[1] - APPLIED COMBINATORICS CHAPTER 1: AN OVERVIEW...

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