On the (upper) line-distinguishing and (upper) harmonious chromatic numbers of a graph

M.Sc.In this dissertation we study two types of colourings, namely line-distinguishing colourings and harmonious colourings. A line-distinguishing colouring of a graph G is a k-colouring of the vertices of G such that no two edges have the same colour. The line-distinguishing chromatic number G is defined as the smallest k such that G has a line-distinguishing k-colouring. A harmonious colouring of a graph G is a proper k-colouring of the vertices of G such that no two edges have the same colour, i.e. no two adjacent vertices can have the same colour. The harmonious chromatic number hG is defined as the smallest k such that G has a line-distinguishing k-colouring. In Chapter 0 we discuss some of the terminology and definitions used later on in our study. In Chapter 1 we introduce line-distinguishing colourings and consider the line-distinguishing chromatic number of certain familiar classes of graphs such as trees, paths, cycles and complete graphs. We also consider graphs with line-distinguishing chromatic number G equal to the usual chromatic number G, and we compare G with the chromatic index G of a graph. In Chapter 2 we mostly discuss minimal line-distinguishing (MLD) colourings. We consider minimal line-distinguishing colourings and graphs of diameter two as well as classes of regular MLD-colourable graphs. We also introduce the concept of distance regular graphs and discuss minimal line-distinguishing colourings in these graphs. In Chapter 3 we introduce a new parameter, namely the upper line-distinguishing chromatic number H G of a graph. We investigate H G for paths and cycles, and consider graphs with small upper line-distinguishing chromatic numbers. In Chapter 4 we consider the second type of colouring studied in this dissertation, namely harmonious colourings. We define the harmonious chromatic number hG and determine hG for certain classes of graphs such as paths, trees, cycles and complete graphs. In Chapter 5 we discuss upper and lower bound for hG. In Chapter 6 we discuss the upper harmonious chromatic number HG of a graph, and we determine HG for paths and cycles. We also consider graphs satisfying HG  G  1. The purpose of this study is to compile a resource which will give a thorough and well-integrated background on line-distinguishing and harmonious colourings. It is also intended to lay the groundwork for further doctoral studies in the field of colourings