Contents Preface Chapter 1 Introduction 1 1.1 Basic concepts 1 1.2 Basic results 5 Chapter 2 Rainbow Connection Number, Order and Minimum Degree (sum) 14 2.1 Rainbow connection number, order and minimum degree (sum) 14 2.2 Rainbow vertex-connection number and minimum degree (sum) 27 Chapter 3 Rainbow Connection Number, Radius and Diameter 38 3.1 Rainbow connection number and radius with bridgeless 38 3.2 Rainbow connection number and radius with bridges 43 Chapter 4 Rainbow Connection Number and Independence Number 52 4.1 Tight example 52 4.2 Lemma and observation 53 4.3 Theorem and proof 54 Chapter 5 Rainbow Connection Number of Dense, Sparse and Random Graphs 69 5.1 Definitions of dense, sparse and random graphs 69 5.2 Rainbow connection number of dense graphs 69 5.3 Rainbow connection number of sparse and random graphs 74 Chapter 6 Computational Complexity 84 6.1 Conjecture of NP-hard and NP-complete 84 6.2 NP-hard and NP-complete 85 Reference 101 Index 106