129 research outputs found

    Chromatic polynomials and network reliability

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    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

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    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

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    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

    Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space

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    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.

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    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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