Thesis (Ph.D.)--University of Western Australia, 2004.

Thesis (M.A.)--University of the Western Cape, 2008. / Includes bibliographical references (leaves 81- 82).

Let K(v,w) denote the complete graph on v vertices with a hole of size w (i.e., K(v, w) = Kv\Kw). We give necessary and sufficient conditions for the existence of a minimum 4-cycle covering of K (v,w).

[Formulae and special characters can only be approximated here. Please see the pdf version of the abstract for an accurate reproduction.] This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph Kn is a partition of its edges into disjoint classes. Each class of edges in a factorisation of Kn corresponds to a spanning subgraph called a factor. If all the factors are isomorphic to one another, then a factorisation of Kn is called an isomorphic factorisation. A homogeneous factorisation of a complete graph is an isomorphic factorisation where there exists a group G which permutes the factors transitively, and a normal subgroup M of G such that each factor is M-vertex-transitive. If M also acts edge-transitively on each factor, then a homogeneous factorisation of Kn is called an edge-transitive homogeneous factorisation. The aim of this thesis is to study edge-transitive homogeneous factorisations of Kn. We achieve a nearly complete explicit classification except for the case where G is an affine 2-homogeneous group of the form ZR p x G0, where G0 is less than or equal to ΓL(1,p to the power of R). In this case, we obtain necessary and sufficient arithmetic conditions on certain parameters for such factorisations to exist, and give a generic construction that specifies the homogeneous factorisation completely, given that the conditions on the parameters hold. Moreover, we give two constructions of infinite families of examples where we specify the parameters explicitly. In the second infinite family, the arc-transitive factors are generalisations of certain arc-transitive, self-complementary graphs constructed by Peisert in 2001.

This thesis treats some of the problems related to the good drawings D$ sb{ rm n}$ of the complete graph K$ sb{ rm n}$. The first of these problems is obtaining all the non-isomorphic good drawings D$ sb{ rm n}$ of K$ sb{ rm n}$. After conjecturing that any good drawing D$ sb{ rm n}$ of K$ sb{ rm n}$ has at least one crossing-free Hamiltonian Circuit, an algorithm generating all the non-isomorphic good drawings D$ sb{ rm n}$ of K$ sb{ rm n}$ is developed. The second problem, determining the existence of a rectilinear drawing D$ sb{ rm n}$ of K$ sb{ rm n}$ with a given set of crossings, is solved by finding a characteristic of the rectilinear drawings D$ sb{ rm n}$ of K$ sb{ rm n}$. An algorithm using this characteristic determines whether a given set of crossing defines a rectilinear drawing D$ sb{ rm n}$ of K$ sb{ rm n}$. The last problem, to generate all the non-isomorphic rectilinear drawings D$ sb{ rm n}$ of K$ sb{ rm n}$, is solved by an algorithm using a set of rectilinear drawings D$ sb{ rm n-1}$ of K$ sb{ rm n-1}$.

Necessary and sufficient conditions are given for the existence of isomorphic decompositions of the complete bipartite graph, the complete graph with a hole, and the λ-fold complete graph into copies of a 4-cycle with a pendant edge.

No description available.

Estudamos sistemas de passeios aleatórios sobre os vértices de um grafo completo. Inicialmente há uma partícula em cada vértice do grafo das quais somente uma está ativa, as outras estão inativas. A partícula ativa realiza um passeio aleatório simples a tempo discreto com tempo de vida que depende do passado do processo, movendo-se ao longo de elos. Quando uma partícula ativa encontra uma inativa, esta se ativa; quando salta sobre um vértice já visitado, morre. O objetivo desta dissertação é estudar a cobertura do grafo completo, ou seja, a proporção de vértices visitados ao fim do processo, quando o número $n$ de vértices tende ao infinito. Analisamos as equações de campo médio para o processo descrito acima, comparando os seus resultados com os do modelo aleatório. Aqui, os resultados do campo médio parecem reproduzir os do modelo aleatório. Depois, apresentamos um estudo similar entre o modelo estocástico e as equações de campo médio para o caso em que cada partícula possui 2 vidas. Finalmente, observamos a cobertura do grafo completo para as equações de campo médio quando o número de vidas por partículas é maior que dois. / We study random walks systems on complete graphs. Initially there is a particle at each vertex of the graph; only one is active and the other are inactive. An active particle performs a discrete-time simple random walk with lifetime depending on the past of the process moving along edges. When an active particle hits an inactive one, the latter is activated. When it jumps on a vertex which has been visited before it dies. The goal of this work is to study the coverage of the complete graph, that is, the proportion of visited vertices at the end of the process, when the number of vertices goes to infinity. We analyze the mean field equations to the process cited above, comparing their results with the ones of the random model. Here the results of the mean field approach seem to reproduce the ones of the random model. After we present a similar study between the stochastic model and mean field approximation to the case that each particle has 2 lifes. Finally we observe the coverage of the complete graph to the mean-field equations when the number of lifes by particle is bigger than two.

Estudamos sistemas de passeios aleatórios sobre os vértices de um grafo completo. Inicialmente há uma partícula em cada vértice do grafo das quais somente uma está ativa, as outras estão inativas. A partícula ativa realiza um passeio aleatório simples a tempo discreto com tempo de vida que depende do passado do processo, movendo-se ao longo de elos. Quando uma partícula ativa encontra uma inativa, esta se ativa; quando salta sobre um vértice já visitado, morre. O objetivo desta dissertação é estudar a cobertura do grafo completo, ou seja, a proporção de vértices visitados ao fim do processo, quando o número $n$ de vértices tende ao infinito. Analisamos as equações de campo médio para o processo descrito acima, comparando os seus resultados com os do modelo aleatório. Aqui, os resultados do campo médio parecem reproduzir os do modelo aleatório. Depois, apresentamos um estudo similar entre o modelo estocástico e as equações de campo médio para o caso em que cada partícula possui 2 vidas. Finalmente, observamos a cobertura do grafo completo para as equações de campo médio quando o número de vidas por partículas é maior que dois. / We study random walks systems on complete graphs. Initially there is a particle at each vertex of the graph; only one is active and the other are inactive. An active particle performs a discrete-time simple random walk with lifetime depending on the past of the process moving along edges. When an active particle hits an inactive one, the latter is activated. When it jumps on a vertex which has been visited before it dies. The goal of this work is to study the coverage of the complete graph, that is, the proportion of visited vertices at the end of the process, when the number of vertices goes to infinity. We analyze the mean field equations to the process cited above, comparing their results with the ones of the random model. Here the results of the mean field approach seem to reproduce the ones of the random model. After we present a similar study between the stochastic model and mean field approximation to the case that each particle has 2 lifes. Finally we observe the coverage of the complete graph to the mean-field equations when the number of lifes by particle is bigger than two.

Cette thèse réunit en trois articles mon intérêt éclectique pour la théorie des graphes. Le premier problème étudié est la conjecture de Erdos-Faber-Lovász: La réunion de k graphes complets distincts, ayant chacun k sommets, qui ont deux-à-deux au plus un sommet en commun peut être proprement coloriée en k couleurs. Cette conjecture se caractérise par le peu de résultats publiés. Nous prouvons qu’une nouvelle classe de graphes, construite de manière inductive, satisfait la conjecture. Le résultat consistera à présenter une classe qui ne présente pas les limitations courantes d’uniformité ou de régularité. Le deuxième problème considère une conjecture concernant la couverture des arêtes d’un graphe: Si G est un graphe simple avec alpha(G) = 2, alors le nombre minimum de cliques nécessaires pour couvrir l’ensemble des arêtes de G (noté ecc(G)) est au plus n, le nombre de sommets de G. La meilleure borne connue satisfaite par ecc(G) pour tous les graphes avec nombre d’indépendance de deux est le minimum de n + delta(G) et 2n − omega(racine (n log n)), où delta(G) est le plus petit nombre de voisins d’un sommet de G. Notre objectif a été d’obtenir la borne ecc(G) <= 3/2 n pour une classe de graphes la plus large possible. Un autre résultat associé à ce problème apporte la preuve de la conjecture pour une classe particulière de graphes: Soit G un graphe simple avec alpha(G) = 2. Si G a une arête dominante uv telle que G \ {u,v} est de diamètre 3, alors ecc(G) <= n. Le troisième problème étudie le jeu de policier et voleur sur un graphe. Presque toutes les études concernent les graphes statiques, et nous souhaitons explorer ce jeu sur les graphes dynamiques, dont les ensembles d’arêtes changent au cours du temps. Nowakowski et Winkler caractérisent les graphes statiques pour lesquels un unique policier peut toujours attraper le voleur, appellés cop-win, à l’aide d’une relation <= définie sur les sommets de ce graphe: Un graphe G est cop-win si et seulement si la relation <= définie sur ses sommets est triviale. Nous adaptons ce théorème aux graphes dynamiques. Notre démarche nous mène à une relation nous permettant de présenter une caractérisation des graphes dynamiques cop-win. Nous donnons ensuite des résultats plus spécifiques aux graphes périodiques. Nous indiquons aussi comment généraliser nos résultats pour k policiers et l voleurs en réduisant ce cas à celui d’un policier unique et un voleur unique. Un algorithme pour décider si, sur un graphe périodique donné, k policiers peuvent capturer l voleurs découle de notre caractérisation. / This thesis represents in three articles my eclectic interest for graph theory. The first problem is the conjecture of Erdos-Faber-Lovász: If k complete graphs, each having k vertices, have the property that every pair of distinct complete graphs have at most one vertex in common, then the vertices of the resulting graph can be properly coloured by using k colours. This conjecture is notable in that only a handful of classes of EFL graphs are proved to satisfy the conjecture. We prove that the Erdos-Faber-Lovász Conjecture holds for a new class of graphs, and we do so by an inductive argument. Furthermore, graphs in this class have no restrictions on the number of complete graphs to which a vertex belongs or on the number of vertices of a certain type that a complete graph must contain. The second problem addresses a conjecture concerning the covering of the edges of a graph: The minimal number of cliques necessary to cover all the edges of a simple graph G is denoted by ecc(G). If alpha(G) = 2, then ecc(G) <= n. The best known bound satisfied by ecc(G) for all the graphs with independence number two is the minimum of n + delta(G) and 2n − omega(square root (n log n)), where delta(G) is the smallest number of neighbours of a vertex in G. In this type of graph, all the vertices at distance two from a given vertex form a clique. Our approach is to extend all of these n cliques in order to cover the maximum possible number of edges. Unfortunately, there are graphs for which it’s impossible to cover all the edges with this method. However, we are able to use this approach to prove a bound of ecc(G) <= 3/2n for some newly studied infinite families of graphs. The third problem addresses the game of Cops and Robbers on a graph. Almost all the articles concern static graphs, and we would like to explore this game on dynamic graphs, the edge sets of which change as a function of time. Nowakowski and Winkler characterize static graphs for which a cop can always catch the robber, called cop-win graphs, by means of a relation <= defined on the vertices of such graphs: A graph G is cop-win if and only if the relation <= defined on its vertices is trivial. We adapt this theorem to dynamic graphs. Our approach leads to a relation, that allows us to present a characterization of cop-win dynamic graphs. We will then give more specific results for periodic graphs, and we will also indicate how to generalize our results to k cops and l robbers by reducing this case to one with a single cop and a single robber. An algorithm to decide whether on a given periodic graph k cops can catch l robbers follows from our characterization.