Combinatorics as a branch of mathematics studies the arts of counting. Enumeration occupies the foundation of combinatorics with a large range of applications not only in mathematics itself but also in many other disciplines. It is too broad a task to write a book to show the deep development in every corner from this aspect. This monograph is intended to provide a unified theory for those related to the enumeration of maps.

 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.

 This book will be of interest to college teachers, graduate students working in mathematics, especially in combinatorics and graph theory, functional and approximate analysis and algebraic systems.

Preface

Chapter 1 Preliminaries

1.1 Maps

1.2 Polynomials on maps

1.3 Enufunctions

1.4 Polysum functions

1.5 The Lagrangian inversion

1.6 The shadow functional

1.7 Asymptotic estimation

1.8 Notes

Chapter 2 0uterplanar Maps

2.1 Plane trees

2.2 Wintersweets

2.3 Unicyclic maps

2.4 General outerplanar maps

2.5 Notes

Chapter 3 Triangulations

3.1 Outerplanar triangulations

3.2 Planar triangulations

3.3 Triangulations on the disc

3.4 Triangulations on the projective plane

3.5 Triangulations on the torus

3.6 Notes

Chapter 4 Cubic Maps

4.1 Planar cubic maps

4.2 Bipartite cubic maps

4.3 Cubic Hamiltonian maps

4.4 Cubic maps on surfaces

4.5 Notes

Chapter 5 Eulerian Maps

5.1 Planar Eulerian maps

5.2 Tutte formula

5.3 Planar Eulerian triangulations

5.4 Regular Eulerian maps

5.5 Notes

Chapter 6 Nonseparable Maps

6.1 Outerplanar nonseparable maps

6.2 Eulerian nonseparable maps

6.3 Planar nonseparable maps

6.4 Nonseparable maps on the surfaces

6.5 Notes

Chapter 7 Simple Maps

7.1 Loopless maps

7.2 Loopless Eulerian maps

7.3 General simple maps

7.4 Simple bipartite maps

7.5 Notes

Chapter 8 General Maps

8.1 General planar maps

8.2 Planar c-nets

8.3 Convex polyhedra

8.4 Quadrangulations via c-nets

8.5 General maps on surfaces

8.6 Notes

Chapter 9 Chrosum Equations

9.1 Tree equations

9.2 Outerplanar equations

9.3 General equations

9.4 Triangulation equations

9.5 Well definedness

9.6 Notes

Chapter 10 Polysum Equations

10.1 Polysum for bitrees

10.2 Outerplanar polysums

10.3 General polysums

10.4 Nonseparable polysums

10.5 Notes

Chapter 11 Chromatic Solutions

11.1 General solutions

11.2 Cubic triangles

11.3 Invariants

11.4 Four color solutions

11.5 Notes

Chapter 12 Stochastic Behaviors

12.1 Asymptotics for outerplanar maps

12.2 The average of tree-rooted maps

12.3 Hamiltonian circuits per map

12.4 The asymmetry of maps

12.5 Asymptotics via equations

12.6 Notes

Bibliography

Index