Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory

The colonisation of the Geographical mind: A critical contextual analysis of the institutionalisation and establishment of Geography as an academic discipline in South Africa

Philosophiae Doctor - PhDThe history of geography is much more than the mere listing of the names and publications of great geographers, identifying different research traditions, or searching for paradigms. The history of geography ought to be seen within the context of the society of which it is an integral part