This monograph is for theory and its extensions related to census of maps on general surfaces established on the basis of what has been done on the surface of genus zero. In spite of a number of improved results with maps on a surface of genus zero for surfaces of genus not zero, two new theoretical lines via exploiting the relationship between super maps and embeddings of a graph on surfaces and via the joint tree technique are investigated for a variety of topics such as those in the determination of handle and crosscap polynomials of maps, genus distribution of embeddings, and others related. In particular, an appendix serves as the exhaustive counting super maps (rooted and nonrooted) including these polynomials with under graphs of small size for the reader's digests.

Since the first monograph titled Enumerative Theory of Maps appeared on the subject considered in 1999, many advances have been made by the author himself and those directed by him under such a theoretical foundation.

Because of that book with much attention to maps on surface of genus zero, this monograph is in principle concerned with maps on surfaces of genus not zero. Via main theoretical lines, this book is divided into four parts except Chapter 1 for preliminaries.

Part one contains Chapters 2 through 10. The theory is presented for maps on general surfaces of genus not necessary to be zero. For the theory on a surface of genus zero is comprehensively improved for investigating maps on all surfaces of genera not zero.

Part two consists of only Chapter 11. Relationships are established for building up a bridge between super maps and embeddings of a graph via their automorphism groups.

Part three consists of Chapters 12 and 13. A general theory for finding genus distribution of graph embeddings, handle polynomials and crosscap polynomials of super maps are constructed on the basis of the joint tree method which enables us to transform a problem in a high dimensional space into a problem on a polygon.

All other chapters, i.e., Chapters 14 through 17, as part four are concerned with several aspects of main extensions to distinct directions.

An appendix serves as atlas of super maps of typical graphs of small size on surfaces for the convenience of readers to check their understanding.

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 Outerplanar 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 Quadrangulations

§4.1 Outerplanar quadrangulations

§4.2 Outerplanar quadrangulations on the disc

§4.3 Hamiltonian quadrangulations on the sphere

§4.4 Inner endless planar quadrangulations

§4.5 Quadrangulations on the projective plane

§4.6 Quadrangulations on the Klein bottle

§4.7 Notes

Chapter 5 Eulerian Maps

§5.1 Planar Eulerian maps

§5.2 Tutte formula

§5.3 Eulerian planar triangulations

§5.4 Regular Eulerian planar maps

§5.5 Eulerian maps on surfaces

§5.6 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 surfaces

§6.5 Bridgeless maps on surfaces

§6.6 Notes

Chapter 7 Simple Maps

§7.1 Loopless maps

§7.2 General simple maps

§7.3 Simple bipartite maps

§7.4 Loopless maps on surfaces

§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 Chrosums on surfaces

§9.7 Notes

Chapter 10 Polysum Equations

§10.1 Polysums for bitrees

§10.2 Outerplanar polysums

§10.3 General polysums

§10.4 Nonseparable polysums

§10.5 Polysums on surfaces

§10.6 Notes

Chapter 11 Maps via Embeddings

§11.1 Automorphism group of a graph

§11.2 Embeddings of a graph

§11.3 Super maps of a graph

§11.4 Maps from embeddings

§11.5 Notes

Chapter 12 Locally Oriented Maps

§12.1 Planar Hamiltonian maps

§12.2 Biboundary inner rooted maps

§12.3 Boundary maps

§12.4 Cubic boundary maps

§12.5 Notes

Chapter 13 Genus Polynomials of Graphs

§13.1 Joint tree model

§13.2 Layer divisions

§13.3 Graphs from smaller

§13.4 Pan-bouquets

§13.5 Notes

Chapter 14 From Rooted to Unrooted

§14.1 Symmetric relations

§14.2 An application

§14.3 Symmetric principles

§14.4 General examples

§14.5 From under graphs

§14.6 Notes

Chapter 15 From Planar to Nonplanar

§15.1 Trees with boundary

§15.2 Cutting along vertices

§15.3 Cutting along faces

§15.4 Maps with a plane base

§15.5 Vertex partition

§15.6 Notes

Chapter 16 Chromatic Solutions

§16.1 General solution

§16.2 Cubic triangles

§16.3 Invariants

§16.4 Four color solutions

§16.5 Notes

Chapter 17 Stochastic Behaviors 

§17.1 Asymptotics for outerplanar maps

§17.2 The average on tree-rooted maps

§17.3 Hamiltonian circuits per map

§17.4 The asymmetry on maps

§17.5 Asymptotics via equations

§17.6 Notes

Appendix  Atlas of Super Maps for Small Graphs

Ax.1 BouquetsBm, 4≥m≥1

Ax.2 Link bundles Lm, 6≥m≥3

Ax.3 Complete bipartite graphs Km,n, 4≥m, n≥3

Ax.4 Wheels Wn, 5≥n≥4

Ax.5 Triconnected cubic graphs of size in [6, 15]

Bibliography

Subject Index

Author Index