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

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

On b-colorings and b-continuity of graphs

A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors. Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs. In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree. At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents 1 Introduction 2 Preliminaries 2.1 Basic terminology 2.2 Colorings of graphs 2.2.1 Vertex colorings 2.2.2 a-colorings 3 b-colorings 3.1 General bounds on the b-chromatic number 3.2 Exact values of the b-chromatic number for special graphs 3.2.1 Graphs with maximum degree at most 2 3.2.2 Graphs with independence number close to its order 3.2.3 Graphs with minimum degree close to its order 3.2.4 Graphs G with independence number plus clique number at most number of vertices 3.2.5 Further known results for special graphs 3.3 Bipartite graphs 3.3.1 General bounds on the b-chromatic number for bipartite graphs 3.3.2 The bicomplement 3.3.3 Bicomplements with simple structure 3.4 Graphs with b-chromatic number close to its t-degree 3.4.1 Regular graphs 3.4.2 Trees and Cacti 3.4.3 Halin graphs 4 b-continuity 4.1 b-spectrum of special graphs 4.2 b-continuous graph classes 4.2.1 Known b-continuous graph classes 4.2.2 Halin graphs 4.3 Further graph properties concerning b-colorings 4.3.1 b-monotonicity 4.3.2 b-perfectness 5 Conclusion Bibliograph

Strong resolvability in product graphs.

En aquesta tesi s'estudia la dimensió mètrica forta de grafs producte. Els resultats més importants de la tesi se centren en la recerca de relacions entre la dimensió mètrica forta de grafs producte i la dels seus factors, juntament amb altres invariants d'aquests factors. Així, s'han estudiat els següents productes de grafs: producte cartesià, producte directe, producte fort, producte lexicogràfic, producte corona, grafs unió, suma cartesiana, i producte arrel, d'ara endavant "grafs producte". Hem obtingut fórmules tancades per la dimensió mètrica forta de diverses famílies no trivials de grafs producte que inclouen, per exemple, grafs bipartits, grafs vèrtexs transitius, grafs hamiltonians, arbres, cicles, grafs complets, etc, i hem donat fites inferiors i superiors generals, expressades en termes d'invariants dels grafs factors, com ara, l'ordre, el nombre d'independència, el nombre de cobriment de vèrtexs, el nombre d'aparellament, la connectivitat algebraica, el nombre de cliqué, i el nombre de cliqué lliure de bessons. També hem descrit algunes classes de grafs producte, on s'assoleixen aquestes fites. És conegut que el problema de trobar la dimensió mètrica forta d'un graf connex es pot transformar en el problema de trobar el nombre de cobriment de vèrtexs de la seva corresponent graf de resolubilitat forta. En aquesta tesi hem aprofitat aquesta eina i hem trobat diverses relacions entre el graf de resolubilitat forta de grafs producte i els grafs de resolubilitat forta dels seus factors. Per exemple, és notable destacar que el graf de resolubilitat forta del producte cartesià de dos grafs és isomorf al producte directe dels grafs de resolubilitat forta dels seus factors.En esta tesis se estudia la dimensión métrica fuerte de grafos producto. Los resultados más importantes de la tesis se centran en la búsqueda de relaciones entre la dimensión métrica fuerte de grafos producto y la de sus factores, junto con otros invariantes de estos factores. Así, se han estudiado los siguientes productos de grafos: producto cartesiano, producto directo, producto fuerte, producto lexicográfico, producto corona, grafos unión, suma cartesiana, y producto raíz, de ahora en adelante "grafos producto". Hemos obtenido fórmulas cerradas para la dimensión métrica fuerte de varias familias no triviales de grafos producto que incluyen, por ejemplo, grafos bipartitos, grafos vértices transitivos, grafos hamiltonianos, árboles, ciclos, grafos completos, etc, y hemos dado cotas inferiores y superiores generales, expresándolas en términos de invariantes de los grafos factores, como por ejemplo, el orden, el número de independencia, el número de cubrimiento de vértices, el número de emparejamiento, la conectividad algebraica, el número de cliqué, y el número de cliqué libre de gemelos. También hemos descrito algunas clases de grafos producto, donde se alcanzan estas cotas. Es conocido que el problema de encontrar la dimensión métrica fuerte de un grafo conexo se puede transformar en el problema de encontrar el número de cubrimiento de vértices de su correspondiente grafo de resolubilidad fuerte. En esta tesis hemos aprovechado esta herramienta y hemos encontrado varias relaciones entre el grafo de resolubilidad fuerte de grafos producto y los grafos de resolubilidad fuerte de sus factores. Por ejemplo, es notable destacar que el grafo de resolubilidad fuerte del producto cartesiano de dos grafos es isomorfo al producto directo de los grafos de resolubilidad fuerte de sus factores.In this thesis we study the strong metric dimension of product graphs. The central results of the thesis are focused on finding relationships between the strong metric dimension of product graphs and that of its factors together with other invariants of these factors. We have studied the following products: Cartesian product graphs, direct product graphs, strong product graphs, lexicographic product graphs, corona product graphs, join graphs, Cartesian sum graphs, and rooted product graphs, from now on ``product graphs''. We have obtained closed formulaes for the strong metric dimension of several nontrivial families of product graphs involving, for instance, bipartite graphs, vertex-transitive graphs, Hamiltonian graphs, trees, cycles, complete graphs, etc., or we have given general lower and upper bounds, and have expressed these in terms of invariants of the factor graphs like, for example, order, independence number, vertex cover number, matching number, algebraic connectivity, clique number, and twin-free clique number. We have also described some classes of product graphs where these bounds are achieved. It is known that the problem of finding the strong metric dimension of a connected graph can be transformed to the problem of finding the vertex cover number of its strong resolving graph. In the thesis we have strongly exploited this tool. We have found several relationships between the strong resolving graph of product graphs and that of its factor graphs. For instance, it is remarkable that the strong resolving graph of the Cartesian product of two graphs is isomorphic to the direct product of the strong resolving graphs of its factors

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view