129 research outputs found
Chromatic polynomials and network reliability
AbstractIn this paper, we introduce and study an extension of the chromatic polynomial of a graph. The new polynomial, determined by a graph G and a subset K of points of G, coincides with the classical chromatic polynomial when K is the set of all points of G. The main theorems in the present paper include analogues of the standard axiomatic characterization and Whitney's topological characterizations of the chromatic polynomial, and the theorem of Stanley relating the chromatic polynomial to the number of acylic of G. The work in this paper was stimulated by important connections between the chromatic polynomial and the all-terminal network reliability problem, and by recent work of Boesch, Satyanarayana, and Suffel on a graph invariant related to the K-terminal reliability problem. Several of the Boesch, Satyanarayana, and Suffel are derived as corollaries to the main theorems of the present paper
Chromaticity of Certain 2-Connected Graphs
Since the introduction of the concepts of chromatically unique graphs and chromatically
equivalent graphs, many families of such graphs have been obtained.
In this thesis, we continue with the search of families of chromatically unique
graphs and chromatically equivalent graphs.
In Chapter 1, we define the concept of graph colouring, the associated chromatic
polynomial and some properties of a chromatic polynomial. We also give some
necessary conditions for graphs that are chromatically unique or chromatically
equivalent.
Chapter 2 deals with the chromatic classes of certain existing 2-connected (n, n + 1,)-graphs for z = 0, 1, 2 and 3. Many families of chromatically unique graphs and
chromatically equivalent graphs of these classes have been obtained. At the end
of the chapter, we re-determine the chromaticity of two families of 2-connected
(n, n + 3)-graphs with at least two triangles. Our main results in this thesis are presented in Chapters 3, 4 and 5. In Chapter
3, we classify all the 2-connected (n, n + 4)-graphs wit h at least four triangles . In
Chapter 4 , we classify all the 2-connected (n, n + 4)-graphs wit h t hree triangles
and one induced 4-cycle. In Chapter 5, we classify all the 2-connected (n, n + 4)graphs
with three triangles and at least two induced 4-cycles . In each chapter, we
obtain new families of chromatically unique graphs and chromatically equivalent
graphs.
We end the thesis by classifying all the 2-connected (n, n + 4)-graphs with exactly
three triangles. We also determine the chromatic polynomial of all these graphs.
The determination of the chromaticity of most classes of these graphs is left as
an open problem for future research
A critically chromatic graph
AbstractDirac (1960) concludes the paper by stating that he does not know whether there exists a critically 4-chromatic graph with connectivity at least 4. Here, we show the existence of such a graph
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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
Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space
For a compact oriented smooth surface immersed in Euclidean four-space (thought of as complex two-space), the sum of the tangential and normal Euler numbers is equal to the algebraic number of points where the tangent plane is a complex line. This follows from the construction of an explicit homology between the zero-chains of complex points and the zero-chains of singular points of projections to lines and hyperplanes representing the tangential and normal Euler classes
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
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Infinite Planar Graphs
How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs.
Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism
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