Author: Liu Yanpei
Language: English
ISBN/ISSN: 7030075978
Published on: 1999-01
Hardcover
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.
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
Language: English
ISBN/ISSN: 7030075978
Published on: 1999-01
Hardcover
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.
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
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