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Aspects of graph colouring
The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below
Towards sustainable use: design behaviour intervention to reduce household environment impact
The use of electrical products has a significant environmental impact, mainly
determined by user behaviour, which has overridden the energy efficiency gains in the
household from technological and educational solutions. Designers are identifiably in a
position to plan and shape the way in which consumption occurs and to fill the gap
between values and everyday user actions. Despite this, the literature demonstrates
that the use phase of the product life cycle is often neglected in sustainable design. Few
attempts have been made to change user behaviour through design-led interventions to
limit its environmental burdens. In addition, there is a lack of understanding of users
perceptions of environmental issues with reference to the specific context: actual use,
habits and fundamental needs of the product as well as the behaviour changing
products. This makes creating sustainable use of the household appliance lessen the
significance of its original purpose.
The aim of this research is to seek the role that design could play in influencing more
sustainable actions to reduce environmental household impacts. Based on a
comprehensive literature review in diverse disciplinary fields of enquiry, a Design
Behaviour Intervention Model has been established to bridge the social-psychological
theories of behaviour and the behaviour intervention approaches. To inform this
enquiry, a single product type (household cold appliance) was chosen as a case study to
explore the capacity of a qualitative behaviour study to identify unsustainable aspects
of product use. Two design activities were carried out: one examining the designer s
ability to respond to the design brief and the other applying the findings that emerged
from the in-depth behaviour analysis and the model into the design process. The
selected outcomes from the design study are evaluated by a focus group to uncover the
users acceptance level of these concepts and the behaviour intervention approaches
applied.
The collective findings are discussed along with the usefulness and effectiveness of the
Design Behaviour Intervention Model in Design for Sustainable Behaviour. This
research highlights that a detailed user study is not only the first step for improving
energy efficiency in product use but also the origin of innovative design concepts to tap
the market by providing better and greener use experiences. Useful insights on primary
findings have emerged: the effectiveness of applying the social-psychological theory in
the Sustainable Design domain; principles of improving effectiveness and acceptability
of the behaviour interventions; and a guide for Design for Sustainable Behaviour