published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

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.

Thesis (Ph. D.)--Lehigh University, 2006. / Includes vita. Includes bibliographical references (leaf 86).

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Iowa, 2007. / Supervisor: Sriram Pemmaraju. Includes bibliographical references (leaves 142-147).

Hajós conjectured that, for any positive integer k, every graph containing no K_(k+1)-subdivision is k-colorable. This is true when k is at most three, and false when k exceeds six. Hajós' conjecture remains open for k=4,5. We will first present some known results on Hajós' conjecture. Then we derive a result on the structure of 2-connected graphs with no cycle through three specified vertices. This result will then be used for the proof of the main result of this thesis. We show that any possible counterexample to Hajós' conjecture for k=4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajós' conjecture for k=4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K_5-subdivision.

Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Yue Zhao. Includes bibliographical references (p. 102-104).

Thesis (M. S.)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Prof. Richard Lipton; Committee Member: Prof. Dana Randall; Committee Member: Prof. H. Venkateswaran. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (M.Sc.). / Title from title page of PDF (viewed 2008/01/30). Written for the School of Computer Science. Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 107-111).