Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

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

Uberising the Urban. Labour, Infrastructure and Big Data in the Actually Existing Smart City of Toronto

This thesis explores how Uber reformats the urban and vice versa. Rather than taking for granted Uber’s success in remoulding the emerging ‘smart city’ in its own image, Uberising the Urban pays close attention to the contradictory, variegated and far from frictionless encounters between Uberisation and urbanisation. The thesis is particularly interested in those neuralgic points of contact where the abstract logics of Uber’s business model – its vectors of data extraction, labour exploitation and platform expansion – hit the urban ground of existing social and physical geographies. The Uberisation of the urban – such is this thesis’s main argument – does not take place in a material and social void; it unfolds in, with and against the dense social and material thickness of existing urban space. This argument is deepened in three case studies. Zooming in from different angles, these case studies show how the vectors of Uberisation have come up against the multiscalar and variously uneven urban grounds of the actually existing smart city of Toronto. While the first case study provides a detailed discussion of the conflictive processes leading up to the legalisation of Uber in Toronto and the parallel ‘regulated deregulation’ of the city’s taxi-cum-ridehail market, the second case study tackles the next subsequent ‘stage’ of Uberisation in Toronto: the proliferation of various public-private ridehail partnerships (PPRPs) between Uber and Lyft on the one hand and local and regional transit agencies in the GTA on the other. The third case study is concerned with Uber’s self-driving car programme and, in particular, the invasive practices of data extraction that Uber has implemented in Toronto – turning the city into a real-life urban data reservoir for the development of its self-driving software. A conclusion, shedding light on a potential reconfiguration of Uber towards more socially emancipatory ends, rounds out the dissertation