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Chapter 1 General Maps 1

1.1 Chromatic Sums of General Maps on the Sphere and the Projective Plane 1

1.1.1 Introduction 1

1.1.2 Maps on the Sphere 2

1.1.3 Maps on the Projective Plane 5

1.1.4 The Case：λ＝∞；λ＝2 9

1.2 Chromatic and Dichromatic Sums of General Maps on the Plane 18

1.2.1 Introduction 18

1.2.2 Chromatic and Dichromatic Sum Equations 19

1.2.3 The Case：λ＝∞；λ＝2 23

1.2.4 Uniqueness of the Solution 25

1.3 Chromatic Sums of General Simple Maps on the Plane 29

1.3.1 Introduction 29

1.3.2 Chromatic Sum Function Equations 30

1.3.3 Uniqueness of the Solution 36

1.3.4 The Case：λ＝∞；λ＝2 40

1.4 Enumeration of Loopless Maps on the Projective Plane 46

1.4.1 Introduction 46

1.4.2 Maps on the Sphere 47

1.4.3 Maps on the Projective Plane 49

1.5 Notes 56

Chapter 2 Nonseparable Maps 59

2.1 Chromatic Sums of Nonseparable Maps on the Sphere and Projective Plane 59

2.1.1 Introduction 59

2.1.2 Maps on the Sphere 61

2.1.3 Maps on the Projective Plane 63

2.1.4 The Case：λ＝∞；λ＝2 66

2.2 Chromatic Sums of Nonseparable Simple Maps on the Plane 75

2.2.1 Introduction 75

2.2.2 Chromatic Sum Function Equations 77

2.2.3 Uniqueness of the Solution 83

2.2.4 The case：λ＝∞；λ＝2 86

2.3 The Number of Nonseparable Map on the Projective Plane 89

2.3.1 Introduction 89

2.3.2 Maps on the Sphere 90

2.3.3 Maps on the Projective Plane 92

2.4 Notes 99

Chapter 3 Singular Maps 100

3.1 Chromatic Sums of Singular Maps on Some Surfaces 100

3.1.1 Introduction 100

3.1.2 Chromatic Sum for Maps on N1 102

3.1.3 Chromatic Sum for Maps on S1 104

3.1.4 Chromatic Sum for Maps on N2 108

3.1.5 The case：λ＝∞ 113

3.2 Singular Maps on the Klein bottle 116

3.2.1 Introduction 116

3.2.2 Enumerating Function with Two Variables 116

3.2.3 Vertex Partition Formula 122

3.3 Bisingular Maps on the Sphere 124

3.4 Bisingular Maps on the Torus 129

3.4.1 Introduction 129

3.4.2 Main Results 130

3.5 Bisingular Maps on the Klein Bottle 137

3.5.1 Introduction 137

3.5.2 Main Results 137

3.6 Notes 148

Chapter 4 Triangular Maps 150

4.1 Chromatic Sums of Rooted Triangulations on the Projective Plane 150

4.1.1 Introduction 150

4.1.2 Main Result 152

4.2 Dual Loopless Nonseparable Near-Triangulations on the Projective Plane 166

4.2.1 Introduction 166

4.2.2 Maps on the Sphere 167

4.2.3 Maps on the Projective Plane 168

4.3 Notes 175

Chapter 5 2-edge-connected Maps 177

5.1 Chromatic Sums of 2-edge-connected Maps on the Plane 177

5.1.1 Introduction 177

5.1.2 Chromatic Sum Equations 179

5.1.3 The case：λ＝∞ 182

5.1.4 The case：λ＝2 187

5.2 Chromatic Sums of 2-edge-connected Maps on the Projective Plane 192

5.2.1 Introduction 192

5.2.2 Chromatic Sum Equations 194

5.2.3 The case：λ＝∞；λ＝2 200

5.3 Notes 203

Bibliography 204