Introductory Map Theory

Kapa & Omega, Glendale, AZ, USA, 2010. - 503 p. Abstract: A Smarandache system (G;R) is such a mathematical system with at least one Smarandachely denied rule r in R such that it behaves in at least two different ways within the same set G, i.e. , validated and invalided, or only invalided but in multiple distinct ways. A map is a 2-cell decomposition of surface, which can be seen as a connected graphs in development from partition to permutation, also a basis for constructing Smarandache systems, particularly, Smarandache 2-manifolds for Smarandache geometry. As an introductory book, this book contains the elementary materials in map theory, including embeddings of a graph, abstract maps, duality, orientable and non-orientable maps, isomorphisms of maps and the enumeration of rooted or unrooted maps, particularly, the joint tree representation of an embedding of a graph on two dimensional manifolds, which enables one to make the complication much simpler on map enumeration. All of these are valuable for researchers and students in combinatorics, graphs and low dimensional topology. Contents: Preface. Chapter I Abstract Embeddings. Chapter II Abstract Maps. Chapter III Duality. Chapter IV Orientability. Chapter V Orientable Maps. Chapter VI Nonorientable Maps. Chapter VII Isomorphisms of Maps. Chapter VIII Asymmetrization. Chapter IX Rooted Petal Bundles. Chapter X Asymmetrized Maps. Chapter XI Maps with Symmetry. Chapter XII Genus Polynomials. Chapter XIII Census with Partitions. Chapter XIV Super Maps of a Graph. Chapter XV Equations with Partitions. Appendix I Concepts of Polyhedra, Surfaces, Embeddings and Maps. Appendix II Table of Genus Polynomials for Embeddings and Maps of Small Size. Appendix III Atlas of Rooted and Unrooted Maps for Small Graphs. Bibliography. Terminology.