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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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Factoring Graphs

Factoring Graphs

(p.125) 7 Factoring Graphs
The Fascinating World of Graph Theory

Arthur Benjamin

Gary Chartrand

Ping Zhang

Princeton University Press

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.

Keywords:   Hall's Theorem, Marriage Theorem, bipartite graph, Tutte's Theorem, Petersen's Theorem, Petersen graph, 1-factorable graph, 1-Factorization Conjecture, 2-factorable graph

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