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The Fascinating World of Graph Theory$
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Ping Zhang, Gary Chartrand, and Arthur Benjamin

Print publication date: 2017

Print ISBN-13: 9780691175638

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691175638.001.0001

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Encircling Graphs

Encircling Graphs

Chapter:
(p.108) 6 Encircling Graphs
Source:
The Fascinating World of Graph Theory
Author(s):

Arthur Benjamin

Gary Chartrand

Ping Zhang

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175638.003.0006

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 + bi (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.

Keywords:   icosian calculus, Hamiltonian graph, Sir William Rowan Hamilton, complex numbers, Icosian Game, Knight's Tour Puzzle, Traveling Salesman Problem, Traveller's Dodecahedron, Voyage Round the World

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