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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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Synchronizing Graphs

Synchronizing Graphs

Chapter:
(p.226) 12 Synchronizing Graphs
Source:
The Fascinating World of Graph Theory
Author(s):

Arthur Benjamin

Gary Chartrand

Ping Zhang

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175638.003.0012

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that the regions of every 3-regular bridgeless planar graph could be colored with four or fewer colors if and only if the edges of such a graph could be colored with three colors so that every two adjacent edges are colored differently. Tait thought that he had found a new way to solve the Four Color Problem. The chapter also examines the chromatic index of a graph, Vizing's Theorem, applications of edge colorings, and a class of numbers in graph theory called Ramsey numbers. Finally, it describes the Road Coloring Theorem which deals with traffic systems consisting only of one-way streets in which the same number of roads leave each location.

Keywords:   edge coloring, Peter Guthrie Tait, planar graph, Four Color Problem, chromatic index, graph, Vizing's Theorem, graph theory, Ramsey number, Road Coloring Theorem

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