The Fascinating World of Graph Theory

The Fascinating World of Graph Theory

Arthur Benjamin
Gary Chartrand
Ping Zhang
Copyright Date: 2015
Pages: 344
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  • Book Info
    The Fascinating World of Graph Theory
    Book Description:

    The fascinating world of graph theory goes back several centuries and revolves around the study of graphs-mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics-and some of its most famous problems. For example, what is the shortest route for a traveling salesman seeking to visit a number of cities in one trip? What is the least number of colors needed to fill in any map so that neighboring regions are always colored differently? Requiring readers to have a math background only up to high school algebra, this book explores the questions and puzzles that have been studied, and often solved, through graph theory. In doing so, the book looks at graph theory's development and the vibrant individuals responsible for the field's growth.

    Introducing graph theory's fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, the Minimum Spanning Tree Problem, the Königsberg Bridge Problem, the Chinese Postman Problem, a Knight's Tour, and the Road Coloring Problem. They present every type of graph imaginable, such as bipartite graphs, Eulerian graphs, the Petersen graph, and trees. Each chapter contains math exercises and problems for readers to savor.

    An eye-opening journey into the world of graphs, this book offers exciting problem-solving possibilities for mathematics and beyond.

    eISBN: 978-1-4008-5200-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-xii)
    A.B., G.C. and P.Z.
  4. Prologue
    (pp. xiii-xvi)

    In a traditional mathematics book, authors typically develop the subject from the bottom up, starting with basic, easier results and gradually leading to more challenging and sophisticated results. This is not what we will do here. Rather, our intention is to display what we consider as some fascinating, beautiful material in an order that we believe will keep the reader interested in the subject and wondering what might lie ahead. Sometimes we’ll prove results, sometimes we won’t. When we don’t prove a result, we’ll supply some intuition to the reader or provide a reference where more information can be found....

  5. 1 Introducing Graphs
    (pp. 1-21)

    The mathematical structure known as agraphhas the valuable feature of helping us to visualize, to analyze, to generalize a situation or problem we may encounter and, in many cases, assisting us to understand it better and possibly find a solution. Let’s begin by seeing how this might happen and what these structures look like.

    We begin with four problems that have a distinct mathematical flavor. Yet any attempt to solve these problems doesn’t appear to use any mathematics you may have previously encountered. However, all of the problems can be analyzed and eventually solved with the aid of...

  6. 2 Classifying Graphs
    (pp. 22-44)

    Many of the topics and problems that we will encounter and that can be represented by graphs deal with the degrees of the vertices of these graphs. Some of these occur in unexpected ways. To see an example of this, we look at a curious and amusing mathematics article that was published more than a quarter of a century ago.

    David Wells, a British mathematician who is the author of many books on mathematics, puzzles, games and mathematics education, has displayed concern over the way mathematics is often taught to high-school students as well as to beginning college students. What...

  7. 3 Analyzing Distance
    (pp. 45-66)

    Distance has been fundamental to civilizations for centuries. Over time many questions involving distance have arisen. What is the distance for a ship to travel between two ports (and how much time would it take a ship to travel that distance)? What is the distance between two cities (by highway or by air)? What is the distance between Earth and Mars? What is the distance for a taxicab to travel between two locations in a major city? For this last question, the distance between two street intersectionsAandBcan be defined as the smallest number of blocks a...

  8. 4 Constructing Trees
    (pp. 67-90)

    In an underdeveloped region of a country, several settlements have grown into villages and the village leaders have decided that it is time to construct paved roads between certain pairs of villages so that it is possible to travel by vehicle between all villages along paved roads. The question is, what is a good way to accomplish this so that the cost involved is kept as low as possible? Figure 4.1a shows a map indicating these villages, which are denoted byv1,v2, … ,v8, along with all practical locations of paved roads and estimated costs (in thousands of...

  9. 5 Traversing Graphs
    (pp. 91-107)

    To pass the time while attending a business meeting or a lecture, some people like to scribble designs with a pencil on a pad of paper (or on an electronic tablet with a drawing pad app). For example, we might place our pencil on the paper (at a point designatedAin Figure 5.1a) and, without lifting our pencil from the paper, arrive at the drawings in Figures 5.1a–d until completing the drawing in Figure 5.1e.

    This rather innocent activity brings up a question that a curious person might ask.

    Given some pencil drawing, can one determine whether it’s...

  10. 6 Encircling Graphs
    (pp. 108-124)

    Inspired by the Königsberg Bridge Problem, the problem of determining conditions under which a graphGhas a circuit containing every edge ofG(necessarily exactly once) was introduced, discussed and solved in the preceding chapter. Under the assumption thatGis connected, such a circuit not only traverses every edge ofGexactly once, it traverses every vertex ofGbut, quite likely, more than once—possibly many times. This brings up the question of when a round-trip can be made in a graph that traverses every vertex of the graph exactly once except, of course, that the terminal...

  11. 7 Factoring Graphs
    (pp. 125-142)

    Suppose that in a certain graduate class in mathematics, there are seven students, namely Alice, Bob, Carla, David, Emma, Frank and Gina, whom we denote bya, b, c, d, e, fandg, respectively. The professor of this class assigns seven challenging problems to them. He tells the class that they can work on each problem in study groups of three students each such that every pair of students belongs only to one of these study groups. Is this even possible? The answer is yes. In fact, the students in the class have divided themselves into the following seven...

  12. 8 Decomposing Graphs
    (pp. 143-163)

    There are many examples of mathematicians who were very young when they made their most famous discoveries. In fact, it is thought by many that the best work of mathematicians occurs during their early years. While this may very well be true of many mathematicians, it is certainly not true of all. One of the major figures in nineteenth-century combinatorics was Thomas Penyngton Kirkman. To many, Kirkman was thought to be an amateur mathematician whose contribution to mathematics consisted of a single problem he invented that dealt with 15 schoolgirls. Kirkman was no amateur mathematician, however. Indeed, he authored some...

  13. 9 Orienting Graphs
    (pp. 164-182)

    While Harvard University is well known for its academic reputation, there have been occasions when it was also known for its athletic achievements. In 1931 Harvard University’s football team was led by all-American quarterback Barry Wood. He was one of the most prominent players of his time and appeared on the cover of the 23 November 1931 issue ofTimemagazine. On 17 October of that year, Harvard played Army and by the end of the first half, Harvard unexpectedly trailed 13–0. During halftime, Harvard President A. Lawrence Lowell, visibly upset, told Lieutenant Colonel (at the time) Robert C....

  14. 10 Drawing Graphs
    (pp. 183-205)

    Decades ago, a puzzle appeared in many books and magazines that has been known by many names. One of the most common names for this puzzle is theThree Houses and Three Utilities Problem.

    Three houses A, B and C are under construction and each house must be provided with connections to each of three utilities, namely water, electricity and natural gas. (See Figure 10.1.) Each utility provider needs a direct line from the utility terminal to each house without passing through another provider’s terminal or another house along the way. Furthermore, all three utility providers need to bury their...

  15. 11 Coloring Graphs
    (pp. 206-225)

    Over the past few centuries, many fascinating mathematics problems have emerged, some quite easy to understand but notoriously difficult to solve.

    One of the famous mathematicians of the seventeenth century was the Frenchman Pierre Fermat. He wrote that for each integern≥ 3, there are no positive integersa, bandcsuch thatan+bn=cn. Of course, there are many positive integer solutions whenn= 2. For example, 32+42= 52, 52+ 122= 132and 82+ 152= 172. A triple (a, b, c) of positive integers such thata2...

  16. 12 Synchronizing Graphs
    (pp. 226-250)

    Among the interests of the Scottish physicist Peter Guthrie Tait (1831–1901) were mathematics and golf. His interest in golf carried over to his son Frederick (better known as Freddie Tait). Indeed, Frederick became the finest amateur golfer of his time.

    Like many others, Peter Tait played a role in the history of the Four Color Problem. In fact, Tait came up with several solutions of the problem himself—unfortunately, all incorrect. One of Tait’s approaches to solve the Four Color Problem was a new idea, one he believed would lead to a different solution. As it turned out, his...

  17. Epilogue Graph Theory: A Look Back—The Road Ahead
    (pp. 251-254)

    Now in its third century, the mathematical area of graph theory had a most humble beginning. The city of Königsberg, located in what was East Prussia in the eighteenth century, became the subject of a question of whether it was possible to stroll about this city and cross each of its seven bridges exactly once. Leonhard Euler, one of the great mathematicians of all time, saw that this problem and a generalization of it could possibly be solved with the aid of a technique called the geometry of position, originated by Gottfried Leibniz, one of the developers of calculus. A...

  18. Exercises
    (pp. 255-308)
  19. Selected References
    (pp. 309-316)
  20. Index of Names
    (pp. 317-318)
  21. Index of Mathematical Terms
    (pp. 319-322)