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Chapter 1 The Background and Review of the Rainbow Connection 1.1 Background and Concepts 1.2 Basic Results Chapter 2 Rainbow Connection Coloring of Edge-version 2.1 Ear Decompositon 2.2 Rainbow Connection Number of a 2-connected Graph 2.3 Hamiltonian Graphs with Maximal Rainbow Connection Number 2.4 Rainbow Connection Number of a 2-edge-connected Graph 2.5 Rainbow 2-edge-connectivity of a 2-edge-connected Graph 2.6 Rainbow 2-edge-connection Number of a 2-edge-connected Graph Chapter 3 Rainbow Vertex-connection of Graphs 3.1 Rainbow Vertex-connection Number of a 2-connected Graph 3.2 Hamiltonian Graphs with Maximal Rainbow Vertex-connection Number Chapter 4 Total Rainbow Connection of Graphs 4.1 Total Rainbow Connection Number of 2-connected Graph 4.2 Some Upper Bounds for the Total Rainbow Connection Number of Graphs Chapter 5 Preliminaries of Zero-sum Problem 5.1 Backgrounds of Zero-sum Problem 5.2 Notation and Definition the Zero-sum Problem 5.3 Overview of the Results of Zero-sum Problems Chapter 6 The Cross Number of Zero-sum Sequences with Some Properites 6.1 Introduction and Main Results 6.2 A Lower Bound for Kl(G) 6.3 Proof of Theorem 6.1 6.4 Concluding Remarks Chapter 7 Two Zero-sum Invariants on Finite Abelian Groups 7.1 Introduction 7.2 Preliminaries 7.3 Proofs of Theorem 7.1 and Corollary 7.1 7.4 Concluding Remarks Chapter 8 Remarks on Tiny Zero-sum sequences 8.1 Introduction 8.2 Preliminaries 8.3 Proof of the Main Results References