Enumerative Theory of Maps

ISBN-13: 9789401058834 / Angielski / Miękka / 2012 / 411 str.

Liu Yanpei

Opis

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. For enumerating maps the first thing we have to know is the sym- metry of a map. Or in other words, we have to know its automorphism group. In general, this is an interesting, complicated, and difficult problem. In order to do this, the first problem we meet is how to make a map considered without symmetry. Since the beginning of sixties when Tutte found a way of rooting on a map, the problem has been solved. This forms the basis of the enumerative theory of maps. As soon as the problem without considering the symmetry is solved for one kind of map, the general problem with symmetry can always, in principle, be solved from what we have known about the automorphism of a polyhedron, a synonym for a map, which can be determined efficiently according to another monograph of the present author [Liu58].

Spis treści

Preface. 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. 2. Outerplanar Maps. 2.1. Plane trees. 2.2. Wintersweets. 2.3. Unicyclic maps. 2.4. General outerplanar maps. 2.5. Notes. 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. 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. 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. 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. 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. 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. 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. 10. Polysum Equations. 10.1. Polysum for bitrees. 10.2. Outerplanar polysums. 10.3. General polysums. 10.4. Nonseparable polysums. 10.5. Notes. 11. Chromatic Solutions. 11.1. General solutions. 11.2. Cubic triangles. 11.3. Invariants. 11.4. Four color solutions. 11.5. Notes. 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.