Encircling Graphs

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Traversing Graphs

This chapter considers Eulerian graphs, a class of graphs named for the Swiss mathematician Leonhard Euler. It begins with a discussion of the the Königsberg Bridge Problem and its connection to Euler, who presented the first solution of the problem in a 1735 paper. Euler showed that it was impossible to stroll through the city of Königsberg, the capital of German East Prussia, and cross each bridge exactly once. He also mentioned in his paper a problem whose solution uses the geometry of position to which Gottfried Leibniz had referred. The chapter concludes with another problem, the Chinese Postman Problem, which deals with minimizing the length of a round-trip that a letter carrier might take.

Analyzing Distance

This chapter considers distance in graphs, first by providing an overview of some fundamental concepts in graph theory. In particular, it discusses connected graphs, cut-vertex and bridge, and bipartite graphs. It then addresses questions of the distance between locations in a graph and those locations that are far from or close to a given location. It also looks at dominating sets in graphs, focusing on the Five Queens Problem/Puzzle and the Lights Out Puzzle, before concluding with an analysis of the rather humorous concept of Erdős numbers, conceptualized by Hungarian mathematician Paul Erdős. According to this concept, for each mathematician A, the Erdős number of A is the distance from A to Erdős in the collaboration graph. Consequently, Erdős is the only mathematician with the Erdős number 0, whereas any mathematician who has coauthored a paper with Erdős has Erdős number 1.

Classifying Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

Graph Theory: A Look Back—The Road Ahead

This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution led to the subject of Eulerian graphs, to the various efforts to solve the Four Color Problem. It considers elements of graph theory found in games and puzzles of the past, and the famous mathematicians involved including Sir William Rowan Hamilton and William Tutte. It also discusses the remarkable increase since the 1960s in the number of mathematicians worldwide devoted to graph theory, along with research journals, books, and monographs that have graph theory as a subject. Finally, it looks at the growth in applications of graph theory dealing with communication and social networks and the Internet in the digital age and the age of technology.

Coloring Graphs

This chapter considers the concept of coloring the vertices of a graph by focusing on the Four Color Problem. It begins with a discussion of three mathematics problems that involve conjecture, attributed to Pierre Fermat, Leonhard Euler, and Christian Goldbach. It then examines one of the most famous problems in mathematics, the Four Color Problem, which addresses the question of whether it is always possible to color the regions of every map with four colors so that neighboring regions are colored differently. After an overview of the origins of the Four Color Problem, the chapter goes on to analyze the Four Color Conjecture, Alfred Bray Kempe's proof of the Four Color Conjecture, and the Five Color Theorem. Finally, it looks at the Four Color Problem in the twentieth century, along with vertex colorings and their applications.

Factoring Graphs

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects can be matched in some way to another collection of objects, with particular emphasis on how different types of schedulings are possible using a graph. It then examines one popular version of Hall's work, a statement known as the Marriage Theorem, the occurrence of matchings in bipartite graphs, Tutte's Theorem, Petersen's Theorem, and the Petersen graph. Peter Christian Julius Petersen introduced the Petersen graph to show that a cubic bridgeless graph need not be 1-factorable. The chapter concludes with an analysis of 1-factorable graphs, the 1-Factorization Conjecture, and 2-factorable graphs.

Introducing Graphs

This chapter provides an introduction to graphs, a mathematical structure for visualizing, analyzing, and generalizing a situation or problem. It first consider four problems that have a distinct mathematical flavor: the Problem of the Five Princes, the Three Houses and Three Utilities Problem, the Three Friends or Three Strangers Problem, and the Job-Hunters Problem. This is followed by discussion of four problems that are not only important in the history of graph theory, but which led to new areas within graph theory: the Königsberg Bridge Problem, the Four Color Problem, the Polyhedron Problem, and the Around the World Problem. The chapter also explores puzzles and problems involving chess that have connections to graph theory before concluding with an overview of the First Theorem of Graph Theory, which is concerned with what happens when the degrees of all vertices of a graph are added.

Decomposing Graphs

This chapter considers problems of whether a graph can be decomposed into certain other kinds of graphs, primarily cycles. It begins with a background on nineteenth-century mathematician Thomas Penyngton Kirkman and the problem he invented known as Kirkman's Schoolgirl Problem, stated as: How many triples can be formed with x symbols in such a way that no pair of symbols occurs more than once in the triple? This is followed by a discussion of the Steiner triple system, the relationship between cyclic decomposition problems and a problem called Alspach's Conjecture, graceful graphs, and the Graceful Tree Conjecture. The chapter concludes with an analysis of the puzzle dubbed Instant Insanity and how graphs can be utilized to solve it.

Constructing Trees

This chapter considers a class of graphs called trees and their construction. Trees are connected graphs containing no cycles. When dealing with trees, a vertex of degree 1 is called a leaf rather than an end-vertex. The chapter first provides an overview of trees and their leaves, along with the relevant theorems, before discussing a tree-counting problem, introduced by British mathematician Arthur Cayley, involving saturated hydrocarbons. It shows that counting the number of saturated hydrocarbons is the same as counting the number of certain kinds of nonisomorphic trees. It then revisits another Cayley problem, one that involved counting labeled trees, and describes Cayley's Tree Formula and the corresponding proof known as the Prüfer code. It also explores decision trees and concludes by looking at the Minimum Spanning Tree Problem and its solution, Kruskal's Algorithm.