内容简介
This monograph provides a unified theory of maps and their enumerations. The crucial idea is to suitably decompose the given set of maps for extracting a functional equation, in order to have advantages for solving or transforming it into those that can be employed to derive as simple a formula as possible. It is shown that the foundation of the theory is for rooted planar maps, since other kinds of maps including nonrooted (or symmetrical) ones and those on general surfaces have been found to have relationships with particular types in planar cases. A number of functional equations and close formulae are discovered in an exact or asymptotic manner. Audience This book will be of interest to college teachers, graduate students working inmathematics, especially in combinatorics and graph theory, functional and approximate analysis and algebraic systems.
目录
Preface Chapter I Preliminaries 11 Maps 12 Polynomials on maps 13 Enufunctions 14 Polysum functions 15 The Lagrangian inversion 16 The shadow functional 17 Asymptotic estimation 18 Notes Chapter 2 Outerplanar Maps 21 Plane trees 22 Wintersweets 23 Unicyclic maps 24 General outerplanar maps 25 Notes Chapter 3 Triangulations 31 Outerplanar triangulations 32 Planar triangulations 33 Triangulations on the disc 34 Triangulations on the projective i 35 Triangulations on the torus 36 Notes Chapter 4 Cubic Maps 41 Planar cubic maps 42 Bipartite cubic maps 43 Cubic Hamiltonian maps 44 Cubic maps on surfaces 45 Notes Chapter 5 Eulerian Maps 51 Planar Eulerian maps 52 Tutte formula 53 Planar Eulerian triangulations 54 Regular Eulerian maps 55 Notes Chapter 6 Nonseparable Maps 61 Outerplanar nonseparable maps 62 Eulerian nonseparable maps 63 Planar nonseparable maps 64 Nonseparable maps on the surfaces 65 Notes Chapter 7 Simple Maps 71 Loopless maps 72 Loopless Eulerian maps 73 General simple maps 74 Simple bipartite maps 75 Notes Chapter 8 General Maps 81 General planar maps 82 Planar c-nets 83 Convex polyhedra 84 Quadrangulations via c-nets 85 General maps on surfaces 86 NotesContents Chapter 9 Chrosum Equations 91 Tree equations 92 Outerplanar equations 93 General equations 94 Triangulation equations 95 Well definedness 96 Notes Chapter 10 Polysum Equations 101 Polysum for bitrees 102 Outerplanar polysums 103 General polysums 104 Nonseparable polysums 105 Notes Chapter II Chromatic Solutions 111 General solutions 112 Cubic triangles 113 Invariants 114 Four color solutions 115 Notes Chapter 12 Stochastic Behaviors 121 Asymptotics for outerplanar maps 122 The average of tree-rooted maps 123 Hamiltonian circuits per map 124 The asymmetry of maps 125 Asymptotics via equations 126 Notes Bibliography Index
