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Edge and total colourings of graphs
Die vorliegenden Arbeit enthĂ€lt Ergebnisse zu Kanten- und TotalfĂ€rbungen von Graphen sowie verschiedenen Variationen dieser FĂ€rbungen. Eine KantenfĂ€rbung eines Graphen G ist eine Zuordnung von Farben zu den Kanten von G, so dass adjazente Kanten unterschiedliche Farben erhalten. Eine TotalfĂ€rbung ist eine FĂ€rbung der Knoten und Kanten von G, so dass adjazente Knoten, adjazente Kanten sowie ein Knoten und eine inzidente Kante jeweils unterschiedlich gefĂ€rbt werden. Der chromatische Index bzw. die totalchromatische Zahl von G bezeichnen die kleinste Anzahl von Farben, mit denen G kantenfĂ€rbbar bzw. totalfĂ€rbbar ist. In dieser Arbeit wird unter anderem die totalchromatische Zahl zirkulanter Graphen mit Maximalgrad 3 bestimmt sowie ein Algorithmus entwickelt, der alle planaren kritischen Graphen der KantenfĂ€rbung mit bis zu 12 Knoten konstruiert und darstellt. Das Konzept der KreisfĂ€rbung von Graphen wird von Knoten- auf Kanten- und TotalfĂ€rbung ĂŒbertragen; Eigenschaften des kreischromatischen Index und der kreistotalchromatischen Zahl werden bewiesen und exakte Werte fĂŒr einige Graphenklassen ermittelt. Die listenchromatische Vermutung wird fĂŒr outerplanare Graphen mit Maximalgrad >4 bewiesen. Die Konzepte der (a,b)- und (a,b,r)-Listen- fĂ€rbung werden von KnotenfĂ€rbung auf KantenfĂ€rbung ĂŒbertragen; es werden Eigenschaften dieser FĂ€rbungen und Ergebnisse fĂŒr einzelne Graphenklassen hergeleitet.This thesis contains results for edge and total colourings as well as for some variations of these colourings. An edge colouring of a graph G is an assignment of colours to the edges of G such that adjacent edges are coloured differently. A total colouring is a colouring of the vertices and edges of G such that adjacent vertices, adjacent edges as well as a vertex and an incident edge are coloured differently. The chromatic index or the total chromatic number of G denote the minimum number of colours such that G admits an edge colouring or a total colouring, respectively. Results in this thesis are - among others - the total chromatic number of circulant graphs with maximum degree 3 and an algorithm to construct and draw all planar critical graphs with at most 12 vertices. The concept of circular colourings is transferred from vertex to edge and total colourings. Properties of the circular chromatic index and the circular total chromatic number are proven and exact values are determined for some classes of graphs. The list chromatic conjecture is confirmed for outerplanar graphs with maximum degree >4; the concepts of (a,b)- and (a,b,r)-list colourings are transferred from vertex to edge colouring and properties of these colourings as well as results for special classes of graphs are given
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
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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 âcolouring differenceâ between two edge-colourings of a graph (Chapter 5) and the âbeta parameterâ (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 Hiltonâs 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
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