# SAMPLES OF HIGHER RATED WRITING: LAB 5

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 This follows from a simple contradiction. If χ(g) = 1, then every vertex is the same colour. However, if G has at least one edge, then at least two vertices with the same colour are connected, and we do not have a k colouring of G. In addition to knowing the minimum value of k needed to have any k-colouring of a graph, it is worthwhile to ask "How many different k-colourings does a graph have for a specific k?" Assuming that k is larger than the chromatic number of the graph being examined (so the amount of possible k-colourings is non-zero), there are a couple of interesting results that come from this question. As it turns out, we can define a function, P G (k), called the chromatic polynomial which counts the number of colourings as a function of k, the number of colours being used. The chromatic polynomial also has a number of properties, and provides at least as much information about a graph as the chromatic number. In fact, the chromatic number is the smallest positive integer that is not a root of the chromatic polynomial; that is, χ(g) is the smallest positive integer for which P G (k) 0. It follows directly from this that every positive integer which is less than the chromatic number must be a factor of the chromatic polynomial (in other words, any positive integer less than the chromatic number is a root of the chromatic polynomial). Additionally, a graph with n vertices will have a chromatic polynomial of degree n, that is, the leading power of k will be k n. It should be noted that, in general, finding the chromatic number and chromatic polynomial for a graph is by no means a simple task (although, it is interesting to note that checking if an approximated value for the chromatic number is correct is much less computationally complex than directly computing the chromatic number). However, if we focus only on special types of graphs, we can bypass this general difficulty and see that certain patterns emerge for these properties in special types of graphs. Applications and Numerical Results Now, returning to the original scenario, it should be easier to see how one might begin examining different camera placement schemes in an art gallery by using graph colouring. As you may have already guessed, the floor plan of the gallery can be modelled using a graph by defining nodes at all corner locations. If we let these nodes represent potential camera placements, then we can use edges to connect any two cameras that are able to view the same region of the gallery, using this as our graph for which we want to find one or more k colourings. If we assume that, from the defined corner positions, each camera can see three directions, then we can look for k colourings using three colours, where each colour represents one possible setup. Then, whichever colour occurs the least amount of times out of all k colourings found will represent the optimal scheme that should be recommended to the museum director to ensure that the entire gallery can be seen by the minimum amount of cameras. In addition to having solved the security camera puzzle, though, the problem of graph colouring can be applied to many other puzzles as well. For example, in a scheduling problem, one could model schedule blocks as nodes and draw lines connecting blocks which cannot happen at the same time, and then look for a k colouring in that graph which will tell them how to formulate the schedule being examined. Graphs can also be used to model networks, geographical maps, and even games! In order to visualise all of this information about graphs and graph colouring, we can use some of the various tools available with Mathematica to create and work with graphs. For example, the following string of commands will give us an image, the chromatic number and polynomial, and a minimal colouring for complete graphs on 3,4,5, and 6 nodes: g1 = CompleteGraph[3]; g2 = CompleteGraph[4]; g3 = CompleteGraph[5]; g4 = CompleteGraph[6]; ShowGraph[g1] ChromaticNumber[g1] ChromaticPolynomial[g1, k] MinimumVertexColoring[g1] ShowGraph[g2] ChromaticNumber[g2] ChromaticPolynomial[g2, k] MinimumVertexColoring[g2] ShowGraph[g3] ChromaticNumber[g3] ChromaticPolynomial[g3, k] MinimumVertexColoring[g3] ShowGraph[g4] ChromaticNumber[g4] ChromaticPolynomial[g4, k] MinimumVertexColoring[g4] Sample 2 2

3 In this article, we explore the proper coloring of the vertices of a finite graph. This notion of graph theory allows for one to color each state of the United States a different color from those of bordering states, while only using four colors (this is from the Four Color Theorem ). Graph coloring has also been applied to many scheduling problems. The Movie Theater Problem To illustrate the usefulness of proper graph coloring, we examine a particular scheduling problem: Suppose you work for a movie theater. You are assigned to minimize the number of show-times while still meeting the demand for each movie. Movies are classified into three categories: low-brow (LB), high-brow (HB), and middle-brow (MB). Demand comes from two demographics: one demographic would like to watch each LB and MB movie, and the other would like to watch each MB and HB movie. For the following movie lineups, how many show-times are needed? (a) 1 LB, 1 MB, 1 HB; (b) 2 LB, 1 MB, 2 HB. To solve these problems, we apply a graphical framework. Represent each movie as a node and demand as edges. For (a), we get: Figure 1 Now apply colors to the nodes such that no two nodes share the same color for any edge, that is, ensure that the edges are properly colored. The number of colors used represents the number of show-times. 3

4 Figure 2 We repeat the process for (b). Figure 3 Here, the movie theater would need three show-times to meet the demand for the five movies. Counting Colors As counting colors has been shown to be very useful, we aim for an algorithm for doing so. Therefore, we need to define the number of colorings. Definition: P (G; x), the number of proper colorings of a graph G using x colors, is the chromatic polynomial of G. 4

5 Counting the number of the proper colorings may be thought of as subtracting the number of improper colorings from the total number of colorings. So, focusing on edge e of Graph H, we remove the edge and examine the resulting graphs. Figure 4 This decomposition is to be repeated until only empty graphs (graphs without edges) remain. To illustrate, we decompose a simpler, three-edge graph. Figure 5 An empty graph with k vertices can be colored in x k ways. So the chromatic polynomial of the above graph is (x 3 x 2 ) (x 2 x), or x 3 2x 2 + x. The smallest nonnegative x for which P (G; x) > 0 is the chromatic number (finding this number would be ideal for the movie theater problem). The algorithm used is known as the Birkhoff-Lewis Algorithm. Sample 3 Graphs are useful tools for representing numerous problems. By developing tools for analyzing graphs, mathematicians can greatly impact a variety of other fields. Examples of problems for which graphs are good representations are making road trips, planning future offerings for flights, and scheduling class offerings. There are several real-world scenarios that graph coloring can be applied to. For example, one of these is figuring out the best sites on the internet. If we make every page on the internet a vertex and draw a line connecting two vertices when the pages are linked, it creates a graph. Google uses this technique to find out which pages are better than others when people use their search engine. A page that is classified as good has many other pages linked to it. Pages that do not receive very many views will not have many edges connected to other vertices. This way, the success of pages can be ranked. This is called PageRank. [1 ] In this report, we will explore how to find chromatic numbers and polynomials of graphs, how to play graph coloring games, and how players can win a vertex coloring game based on a graph s chromatic number and the number of colors that are used to color it. TERMINOLOGY Graph: A graph is a collection of vertices and edges between the vertices. Connected: A graph is connected if for any two vertices there is an edge-path from one to the other. Component: Vertices are a component if they are connected to another vertex. Proper Vertex Coloring: A vertex coloring of a graph is proper if no two vertices sharing an edge have 5

6 the same color. Path: A path is an order pairing of edges such that each sequential edge is connected to the next edge in the pairing. Hamiltonian path: A Hamiltonian path is a path that visits every vertex once. Euler path: An Euler path is a path that visits every edge exactly once. Polynomial time problem: A polynomial run time problem is a problem for which the number of operations required to find a solution can be represented using only polynomial terms. Many difficult problems have no known polynomial time solutions, but such solutions are highly desirable. Chromatic number: The smallest number of colors needed to color a graph with no two adjacent vertices sharing the same color. Chromatic Polynomial: A chromatic polynomial is a polynomial generated from a graph G such that the polynomial will produce the number of proper colorings for an inputted number of colors x. This is denoted p(g, x). TECHNICAL FINDINGS Color-Assigning Method We found that for most graphs the chromatic polynomial could be found by choosing a starting vertex, and designating this vertex to any of the available colors. Then we choose an adjacent vertex that has not yet been assigned, and assign it to any of the available colors where an available color is a choice that keeps the coloring of the graph proper. If a vertex is connected to k vertices that have already been assigned a color, this vertex is assigned the value (x k). If there are no adjacent vertices that have not been assigned a value, jump to another vertex that is connected to a vertex that has already been assigned. If no unassigned vertices are adjacent to previously assigned vertices, choose one of the unassigned vertices, and start the assignment process over again. Once all vertices have been assigned a value, the chromatic polynomial is all of the values of the vertices multiplied together. Conjecture 1. The Color-Assigning Method produces the chromatic polynomial when applied to a graph G. Proof For a vertex connected to k vertices that have already been assigned a color, assigning the value (x k) is representative of the number of colors that vertex could be colored. In order for the coloring to remain proper, no connected vertices may be the same color. Thus any color may be assigned to that vertex other than the color of the two other vertices, leaving the coloring choices (x k). The chromatic polynomial of the graph is represented by the product of all of the assignments because the number of combinations of colors is equal to the product of the number of colors available at each vertex. Birkhoff-Lewis Counting Algorithm The Birkhoff-Lewis Counting Algorithm is an algorithm that, given the graph g, produces the chromatic polynomial for g. The algorithm works by creating two new graphs from the original graph. One of the graphs has an edge from connected vertices v 1 and v 2 removed, while the other has combined v 1 and v 2 into a new vertex v 3 such that v 3 has all of the connections that v 1 and v 2 previously had. The chromatic polynomial for the original graph is equal to the chromatic polynomial of the graph with the edge between v 1 and v 2 removed minus the chromatic polynomial of the graph with the vertices merged. By repeating this process, the chromatic polynomial can be found as a combination of the chromatic polynomials for graphs with no connections, and graphs with a single vertex. Conjecture 2. The chromatic polynomial for a graph which contains a single unconnected vertex is x. Proof With a single vertex and x colors, each of the colors may be used to color the vertex, thus the number of proper colors is equal to the number of colors available. Thus the chromatic polynomial is x. Conjecture 3. The chromatic polynomial for a graph with n vertices and no connections is x n. Proof The number of color choices available for a vertex is dependent on the number of edges that vertex has to other vertices. If a vertex has no connections, there are no possible color conflicts, so all of the x considered colors are valid choices. With n coloring spots and x valid colors for each spot, the chromatic polynomial becomes x n. 6

7 4. Technical Findings Since a complete graph will have every node connected to every other node, it would only make sense that the chromatic number for a complete graph on 2, 3, 4, 5, and 6 vertices will be 2, 3, 4, 5, and 6 respectively. This is the case because no two vertices can have the same color in a complete graph. For example, the following is a complete graph on 6 vertices: Clearly, the graph shows that the chromatic number has to be 6 because no two vertices can share the same color Question 2. Sample 1 to every other vertex. A simple example to imagine would be to think of a triangle, all 3 vertexes connect to the remaining 2 vertexes. Now consider a graph of n nodes that is complete. We can ask how many colors does it take to color that graph. We can begin by looking at 1, 2, 3 and 4 These correspond to a point, a line, a triangle, and a square seen below. Since each of the points connects to each of the other points so a 1 node graph needs 1 color, 2 nodes needs 2 all the way to 4 nodes needs 4. This gives us a hint that we need to do a proof by induction to prove how many colors is needed for a completed graph of n nodes. First lets prove that 1 S since that is a graph of 1 node it only needs 1 color to be colored. Next we need to prove that if an n node graph needs n colors then an n+1 node graph needs n+1 colors. Assume that n S then the n node graph has an n chromatic color. Now consider that graph with an added node. To properly color this node it has to be separate from each other node. Since each other node has a unique color since the graph is complete it must be a new color making the chromatic color of the n + 1 node graph n + 1, so n + 1 S so S = N Question 5. Sample 1 H: A graph with 6 vertices and 9 edges. Chromatic polynomial: x 6 9x x 4 75x x 2 31x = (x 1) x (x 4 8x x 2 47x + 31) Chromatic number: I: A graph with 6 vertices and 5 edges. Chromatic polynomial: x 6 5x x 4 10x 3 + 5x 2 x = x (x 1) 5 Chromatic number: 2 7

8 J: A graph with 7 vertices and 11 edges. Chromatic polynomial: x 7 11x 6 +50x 5 121x x 3 120x 2 +36x = (x 3) (x 1) x (x 2) 2 (x 2 3x+3) Chromatic number: K: A graph with 8 vertices and 10 edges. Chromatic polynomial: x 8 10x 7 +43x 6 103x x 4 127x 3 +60x 2 12x = x (x 2) 2 (x 1) 3 (x 2 3x+3) Chromatic number: Sample 2 Our next investigation was to use sagemath.org to find the chromatic polynomials for each of the graphs in figure below. The error Graph is not defined suggest that the graph class is not part of the online library. Therefore, the next step was to find it by hand. This involves picking a vertex, assigning x possible colors to it, and then picking another. If it is connected to the first, that color can t be used. If it is connected to two, then we need to know if the two are different or the same to assign a subtraction value. The in class examples showed that if a vertex s color options can t be determined, its corresponding factors are an unfactorable polynomial of degree 2. Graph H s is x(x 1)(x 1)(unfactorable polynomial)(unfactorable polynomial)(unfactorable polynomial). The last 3 terms are the result of not knowing how the non-adjacent vertices relate to teach other, so that we cannot assign x some number of color options removed. Since (x 1) is the smallest known term, we tried x = 2 first, which worked. Graph I s is x(x 1) (x 1) (x 1) (x 1) (x 1). Clearly the center can have any value, and the outside ones cannot have that value. Only 2 colors are needed. Graph J s is x(x 1)(x 1)(x 2)(x 2)(x - 3)(unfactorable), with the last factor depending on non-adjacent vertices of unknown relationships. Therefore, x = 4 is a likely first guess. Note that J contains the 4 vertex complete graph, a square with opposite vertices connected. See the figure, which uses 4 colors. Graph K s chromatic polynomial is x(x 1)(x 1)(x 1)(x 1)(x 2)(unfactorable)(unfactorable). This means at least 3 colors are needed, but maybe more are needed, based on the last term being unknown. Only 3 were indeed needed, in the figure. 8

9 Graph H: Sample 3 A proper coloring that we found for this graph was [[2, 4, 6], [1, 3, 5]] where the each grouping is vertices of the same color. The chromatic number for this graph is 2. 9

10 Graph I: A proper coloring that we found for this graph was [[6], [1, 2, 3, 4, 5]] where the each grouping is vertices of the same color. The chromatic number for this graph is 2. Graph J: A proper coloring that we found for this graph was [[6, 2], [7], [1, 5], [4, 3]] where the each grouping is vertices of the same color. The chromatic number for this graph is 4. Graph K: 10

11 A proper coloring that we found for this graph was [[5, 3], [1, 2, 6, 8], [4, 7]] where the each grouping is vertices of the same color.the chromatic number for this graph is Question 10. Sample 1 If the chromatic number of a graph G is k (for some positive integer k), what factors do you know must exist for the chromatic polynomials of G? If the chromatic number of graph G is k then the outcome of the smallest factor of the chromatic polynomial of G must equal 1. These factors must be present: (x)(x 1)(x 2)...(x (k 1)). Why is this so? Recall that the least number of needed colors to color a graph is k, by definition of the chromatic number. Therefore, all numbers a such that 0 a < k must fail to create any valid color combinations: In other words, the chromatic polynomial must equal 0 when x = a for the a values described above. To ensure this happens, the factors (x)(x 1)(x 2)...(x (k 1)) must be present in the factorization of the chromatic polynomial. Side note: Recall that the chromatic polynomial of a complete graph of vertex count k has this as its exact polynomial. The connection is this: A complete graph of vertex count k needs exactly k colors to abide by the rules of graph coloring, no more and no less Question 12. Sample 1 Looking at the complete graphs we can see that the expanded polynomial is interesting. For this we are going to take a 10 node completed graph and remove one edge from the same node until it is separate from the graph. When we did this we were returned the following chromatic polynomial 0 edged removed x^10-45*x^ *x^8-9450*x^ *x^ *x^ *x^ *x^ *x^ *x 1 edged removed x^10-44*x^ *x^8-8904*x^ *x^ *x^ *x^ *x^ *x^ *x 2 edged removed x^10-43*x^ *x^8-8358*x^ *x^ *x^ *x^ *x^ *x^ *x 3 edged removed x^10-42*x^ *x^8-7812*x^ *x^ *x^ *x^ *x^ *x^ *x 4 edged removed x^10-41*x^ *x^8-7266*x^ *x^ *x^ *x^ *x^ *x^ *x 5 edged removed x^10-40*x^ *x^8-6720*x^ *x^ *x^ *x^ *x^ *x^ *x 6 edged removed 11

12 x^10-39*x^ *x^8-6174*x^ *x^ *x^ *x^ *x^ *x^ *x 7 edged removed x^10-38*x^ *x^8-5628*x^ *x^ *x^ *x^ *x^ *x^ *x 8 edged removed x^10-37*x^ *x^8-5082*x^ *x^ *x^ *x^ *x^ *x^ *x 9 edged removed x^10-36*x^ *x^8-4536*x^ *x^ *x^ *x^ *x^ *x^2 Looking at the coefficient associated with each polynomial we get a pattern of subtraction from them. The table below shows the power of x and the difference each edge creates in that coefficient. The completed graph x power difference This seems to indicate that removing one from each will eventually remove the x 1 term which makes intuitive sense because when that is completely separate it would be a free coloring. Sample 2 For this question, we talk about chromatic polynomials of the form: a n x n + a n 1 x n 1 + a n 2 x n 2... a 2 x 2 + a 1 x In all of the graphs we looked at we noted that: a n = 1 a n 1 = E where E is the number of edges n i=1 (a i) = 0 We believe that there are more connections, yet due to the limited scope of our study, the above 3 were the only few that arose. Sample 3 For this question, we decided to look at n-cycle graphs, because they were the one exception to the trend that we previously found for finding the chromatic number of a graph. To begin we generated n-cycle graphs for n = 3 6, and found their chromatic polynomials. Graph A: 12

13 The chromatic polynomial for this graph is x 3 3x 2 + 2x which factors into (x 2)(x 1)(x). 13

14 Graph B: The chromatic polynomial for this graph is x 4 4x 3 + 6x 2 3x which factors into (x 2 3x + 3)(x 1)(x). Graph C: The chromatic polynomial for this graph is x 5 5x x 3 10x 2 + 4x which factors into (x 2)(x 1)(x)(x 2 2x + 2). Graph D: 14

15 The chromatic polynomial for this graph is x 6 6x x 4 20x x 2 5x which factors into (x 1)(x)(x 4 5x x 2 10x + 5). From these graphs and their chromatic polynomials we noticed a fascinating pattern. on the chromatic expressions are the binomial coefficients but slightly altered. The coefficients Conjecture 4. For an n-cycle graph, the coefficients of the chromatic polynomial are equal to the coefficients of the n th layer of Pascal s triangle except the last coefficient of the chromatic polynomial is equal to one less than the second to last number on that layer of Pascal s triangle. We found this pattern to be incredible because it suggests a connection between these types of graphs and the pascals triangle. We were not able to find a proof to support our conjecture, but we included it because we feel that it suggests something fundamentally interesting. ( It is amazing how everything ends up connected!) 4.5. Question 13. Here we record only the various ways in which the game was defined in the lab reports submitted. Sample 1 Two players, Q (P1) and M (P2) take turns colouring vertexes of a random graph. If the graph is successfully (properly) coloured at the end of the game, then Q wins, but if the game reaches a point where a vertex v cannot be properly coloured (it is connected to another vertex of every available colour), then M wins the game. Sample 2 A game can be made from coloring graphs, the rules are as follows: First, this is a two player, perfect information game. Player one goes first (referred to from now on as P1) and alternates turns with player two (referred to as P2) to color the vertexes of a graph so that no vertexes sharing an edge have the same color. A coloring of a graph that uses at most k colorings is called a proper k coloring. The smallest number of colors needed to color a graph is its chromatic number and is denoted X(G). Sample 3 Rules: Draw any number of vertices and connect them any way you want. Player ONE and player TWO color vertices of a graph with k colors. Player ONE will make a move by coloring a vertex with a certain color. Player TWO then colors a different vertex with their own color, but there cannot be two vertices of the same color that share an edge. A player wins when they arrive at a correct coloring of the graph by coloring the last vertex, or if a player forces the other to violate the proper coloring condition by putting them in a position where they must color a vertex whose adjacent vertices are the same color. Sample 4 Two players may pick from a set of k colors to color a graph, taking turns. Player ONE wishes to color the graph with no adjacent vertices having the same color, whereas Player TWO wishes to color vertices such that this becomes impossible to do with only the k colors. Sample 5 PLAYER ONE and PLAYER TWO are coloring the vertices of a graph G with a set k of colors. 15

16 PLAYER ONE and PLAYER TWO are taking turns, coloring properly a uncolored vertex (in the standard version, PLAYER ONE begins). If a vertex v is impossible to color properly (for any color, v has a neighbor colored with it), then PLAYER TWO wins. If the graph is completely colored, then PLAYER ONE wins. 5. Conclusion Sample 1 To conclude, we have defined the basic notion of a graph, discussed the basic components of graph colouring, and looked at some applications and results from the theoretical discussion. Graph colouring can be used in numerous applications, from setting up security cameras to scheduling to modelling a network. Colouring graphs is not a trivial task, but if the graph we want to colour is a special type of graph (like a tree), then we can look to patterns in the chromatic polynomial to tell us more about the graph. Continuing on, it seems worthwhile to begin seriously investigating computationally efficient methods for finding the chromatic polynomial and chromatic number for large graphs. While it is always possible that the end result will merely prove that there is no algorithm more computationally efficient than what we already have for graph colouring, the enormous benefit of finding such an algorithm, if it does exist, certainly seems worth the risk. 16

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### Sudoku puzzles and how to solve them

Sudoku puzzles and how to solve them Andries E. Brouwer 2006-05-31 1 Sudoku Figure 1: Two puzzles the second one is difficult A Sudoku puzzle (of classical type ) consists of a 9-by-9 matrix partitioned

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### 1. Write the number of the left-hand item next to the item on the right that corresponds to it.

1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Stanford prison experiment 2. Friendster 3. neuron 4. router 5. tipping 6. small worlds 7. job-hunting

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### C H A P T E R Regular Expressions regular expression

7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

### 1 Introduction to Counting

1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Introduction to computer science

Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical

### Planar Graphs. Complement to Chapter 2, The Villas of the Bellevue

Planar Graphs Complement to Chapter 2, The Villas of the Bellevue In the chapter The Villas of the Bellevue, Manori gives Courtel the following definition. Definition A graph is planar if it can be drawn

### Introduction to Diophantine Equations

Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

### Permutation Groups. Rubik s Cube

Permutation Groups and Rubik s Cube Tom Davis tomrdavis@earthlink.net May 6, 2000 Abstract In this paper we ll discuss permutations (rearrangements of objects), how to combine them, and how to construct

### The Taxman Game. Robert K. Moniot September 5, 2003

The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Continuous Functions, Smooth Functions and the Derivative

UCSC AMS/ECON 11A Supplemental Notes # 4 Continuous Functions, Smooth Functions and the Derivative c 2004 Yonatan Katznelson 1. Continuous functions One of the things that economists like to do with mathematical

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### SMMG December 2 nd, 2006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. Fun Fibonacci Facts

SMMG December nd, 006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. The Fibonacci Numbers Fun Fibonacci Facts Examine the following sequence of natural numbers.

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### Pick s Theorem. Tom Davis Oct 27, 2003

Part I Examples Pick s Theorem Tom Davis tomrdavis@earthlink.net Oct 27, 2003 Pick s Theorem provides a method to calculate the area of simple polygons whose vertices lie on lattice points points with

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours Essential Question: LESSON 4 FINITE ARITHMETIC SERIES AND RELATIONSHIP TO QUADRATIC

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### NODAL ANALYSIS. Circuits Nodal Analysis 1 M H Miller

NODAL ANALYSIS A branch of an electric circuit is a connection between two points in the circuit. In general a simple wire connection, i.e., a 'short-circuit', is not considered a branch since it is known

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### This puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children.

0.1 Friend Trends This puzzle is based on the following anecdote concerning a Hungarian sociologist and his observations of circles of friends among children. In the 1950s, a Hungarian sociologist S. Szalai

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### SMOOTH CHESS DAVID I. SPIVAK

SMOOTH CHESS DAVID I. SPIVAK Contents 1. Introduction 1 2. Chess as a board game 2 3. Smooth Chess as a smoothing of chess 4 3.7. Rules of smooth chess 7 4. Analysis of Smooth Chess 7 5. Examples 7 6.

### Unit 12: Introduction to Factoring. Learning Objectives 12.2

Unit 1 Table of Contents Unit 1: Introduction to Factoring Learning Objectives 1. Instructor Notes The Mathematics of Factoring Teaching Tips: Challenges and Approaches Additional Resources Instructor

### LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton

LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton 1. Take a two-digit number and write it down three times to form a six-digit number. For example,

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### Intersecting Two Lines, Part One

Module 1.4 Page 97 of 938. Module 1.4: Intersecting Two Lines, Part One This module will explain to you several common methods used for intersecting two lines. By this, we mean finding the point x, y)

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### 1. The Fly In The Ointment

Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### Sequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium

Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### GRAPHS AND ZERO-DIVISORS. In an algebra class, one uses the zero-factor property to solve polynomial equations.

GRAPHS AND ZERO-DIVISORS M. AXTELL AND J. STICKLES In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Chapter 7. Hierarchical cluster analysis. Contents 7-1

7-1 Chapter 7 Hierarchical cluster analysis In Part 2 (Chapters 4 to 6) we defined several different ways of measuring distance (or dissimilarity as the case may be) between the rows or between the columns

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides