Traversing Graphs

The Fascinating World of Graph Theory ◽

10.23943/princeton/9780691175638.003.0005 ◽

2017 ◽

Author(s):

Arthur Benjamin ◽

Gary Chartrand ◽

Ping Zhang

Keyword(s):

Chinese Postman Problem ◽

Round Trip ◽

Eulerian Graphs ◽

Bridge Problem ◽

Chinese Postman ◽

Leonhard Euler ◽

The City ◽

East Prussia

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.

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On known and less known relations of Leonhard Euler with Poland

Studia Historiae Scientiarum ◽

10.4467/23921749shs.16.005.6148 ◽

2016 ◽

Vol 15 ◽

pp. 75-110

Author(s):

Roman Sznajder ◽

Keyword(s):

Graph Theory ◽

18Th Century ◽

Mathematical Community ◽

Geometric Interpretation ◽

European Countries ◽

Complex Numbers ◽

And Topology ◽

Bridge Problem ◽

Leonhard Euler ◽

The City

In this work we focus on research contacts of Leonhard Euler with Polish scientists of his era, mainly with those from the city of Gdańsk (then Gedanum, Danzig). L. Euler was the most prolific mathematician of all times, the most outstanding mathematician of the 18th century, and one of the best ever. The complete edition of his manuscripts is still in process (Kleinert 2015; Kleinert, Mattmüller 2007). Euler’s contacts with French, German, Russian, and Swiss scientists have been widely known, while relations with Poland, then one of the largest European countries, are still in oblivion. Euler visited Poland only once, in June of 1766, on his way back from Berlin to St. Petersburg. He was hosted for ten days in Warsaw by Stanisław II August Poniatowski, the last king of Poland. Many Polish scientists were introduced to Euler, not only from mathematical circles, but also astronomers and geographers. The correspondence of Euler with Gdańsk scientists and officials, including Carl L. Ehler, Heinrich Kühn and Nathanael M. von Wolf, originated already in the mid-1730s. We highlight the relations of L. Euler with H. Kühn, a professor of mathematics at the Danzig Academic Gymnasium and arguably the best Polish mathematician of his era. It was H. Kühn from whom Euler learned about the Königsberg Bridge Problem; hence one can argue that the beginning of the graph theory and topology of the plane originated in Gdańsk. In addition, H. Kühn was the first mathematician who proposed a geometric interpretation of complex numbers, the theme very much appreciated by Euler. Findings included in this paper are either unknown or little known to a general mathematical community.

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Graph Theory: A Look Back—The Road Ahead

The Fascinating World of Graph Theory ◽

10.23943/princeton/9780691175638.003.0013 ◽

2017 ◽

Author(s):

Arthur Benjamin ◽

Gary Chartrand ◽

Ping Zhang

Keyword(s):

Social Networks ◽

Graph Theory ◽

The Internet ◽

The Road ◽

Eulerian Graphs ◽

Bridge Problem ◽

Look Back ◽

Leonhard Euler ◽

The Subject ◽

The 1960S

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.

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