Contents Chapter 1 Acyclic Coloring 1 1.1 Basic Definitions and Notations 1 1.2 Acyclic Vertex Coloring 1 1.3 Generalized Acyclic Vertex Coloring 10 1.3.1 r-acyclic Vertex Coloring 11 1.3.2 Degenerating Coloring 17 1.4 Acyclic Edge Coloring 29 1.5 Open Problems 47 Reference 47 Chapter 2 Neighbor Sum Distinguishing Coloring 51 2.1 Basic Definitions and Introduction 51 2.2 Neighbor Sum Distinguishing Edge Coloring of Graphs 52 2.2.1 The Conjecture and Related Results 52 2.2.2 The List Version of Neighbor Sum Distinguishing Edge Colorings 75 2.3 Neighbor Sum Distinguishing Total Coloring of Graphs 76 2.3.1 The Conjecture and Related Results 76 2.3.2 Neighbor Sum Distinguishing Total Choosability of Graphs 78 2.4 Equitable Neighbor Sum Distinguishing Coloring 86 Reference 87 Chapter 3 Edge Cover Coloring 91 3.1 The Edge Cover Coloring of Graphs 91 3.1.1 Edge Cover Chromatic Index and Classi-cation of Graphs 91 3.1.2 Edge Covered Critical Graphs 97 3.1.3 Unsolved Problems on Edge Cover Coloring 102 3.2 Fractional Edge Cover Coloring of Graphs 102 3.2.1 Introduction 103 3.2.2 Main Results on Fractional Edge Cover Coloring of Graphs 105 3.2.3 The Conjectures and Discussions on Fractional Edge Cover Coloring 109 3.3 g-edge Cover Coloring of Graphs 110 3.3.1 g-edge Cover Chromatic Index of Graphs 111 3.3.2 g-Edge Covered Critical Graphs 113 3.3.3 Related Problems on g-edge Cover Coloring of Graphs 121 Reference 122 Chapter 4 f-colorings of Graphs 124 4.1 Introduction 124 4.2 Basic Definitions and Tools 125 4.3 f-colorings of Multigraphs 127 4.4 The Classification Problem of Simple Graphs on f-Colorings 131 4.4.1 Main Results 133 4.4.2 Application in Proper Edge Colorings of Simple Graphs 140 4.4.3 Further Discussion 143 4.5 Critical Graphs on f-colorings 144 4.5.1 Some Properties of f-critical Graphs 145 4.5.2 Bounds on the Number of Edges of f-critical Graphs 148 4.5.3 f-regular f-critical Graphs 151 4.5.4 f-critical Graphs with The f-core Having Maximum Degree 2 153 4.5.5 Some problems for future research 159 4.6 Equitable Edge-colorings of Simple Graphs 160 4.6.1 A Useful Lemma 163 4.6.2 A Problem for Further Research 184 Reference 185 Chapter 5 Total Coloring 188 5.1 Introduction 188 5.2 Total Coloring Conjecture and the Related Results on *≥9 188 5.2.1 Total Coloring Conjecture 188 5.3 Total Coloring of Graph G with *(G) ≤8 202 5.3.1 The Results on the Case*≤8 202 5.3.2 The Results on the Case*≤7 220 5.4 List Total Coloring of Graphs 234 5.5 The Open Problems and Conjectures 243 Reference 243 Index 247
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