We first prove a ``non-embeddable extensions' theorem for polyhedral graph embeddings. Let G be a ``weakly 4-connected' planar graph. We describe a set of constructions that produce a finite list of non-planar graphs, each having a minor isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. The theorem is more general and applies in particular to polyhedral embeddings in any surface. We discuss an approach to proving Jorgensen's conjecture, which states that if G is a 6-connected graph with no K_6 minor, then it is apex, that is, it has a vertex v such that deleting v yields a planar graph. We relax the condition of 6-connectivity, and prove Jorgensen's conjecture for a certain sub-class of these graphs. We prove that every graph embedded in the Klein bottle with representativity at least 4 has a K_6 minor. Also, we prove that every ``locally 5-connected' triangulation of the torus, with one exception, has a K_6 minor. (Local 5-connectivity is a natural notion of local connectivity for a surface embedding.) The above theorem uses a locally 5-connected version of the well-known splitter theorem for triangulations of any surface. We conclude with a theoretically optimal algorithm for the following graph connectivity problem. A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3-shredder is a shredder of size three. We present an algorithm that, given a 3-connected graph, finds its 3-shredders in time proportional to the number of vertices and edges, when implemented on a RAM (random access machine).

We determine necessary and sufficient conditions for a Cayley digraph of the cyclic group of order n to have the property that any other Cayley digraph of a cyclic group of order n is isomorphic to the first if and only if an isomorphism between the two digraphs is a group automorphism of the cyclic group of order n.

The following new results concerning 1-factorizations of the complete graph are proved: (1) There are exactly 6 equivalence classes of 1-factorizations of the complete graph with 8 vertices. (2). There are exactly 396 equivalence classes of 1-factorizations of the complete graph with 10 vertices. Representatives of each of the equivalence classes are presented. The size of the automorphism group of each equivalence class of 1-factorizations of the complete graph with 2n vertices for n ≤ 5 is also found. Several theorems and results related to 1-factorizations of the complete graph are presented, and the relationship to round robin schedules is shown. An application problem demonstrates the importance of the choice of the equivalence class of round robin schedules in the solvability of the problem. / Graduate

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

No description available.

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.

O objetivo deste trabalho é introduzir a noção de grafos familiarizando os alunos com um conceito pouco estudado no ensino fundamental e médio. Para isso, foram estudados algumas situações práticas e a resolução por meio de grafos. A apresentação da teoria de grafos é feita utilizando alguns dos problemas clássicos (Pontes de Königsberg e o Problema do caixeiro-viajante) que originaram a teoria tal como é conhecida nos dias de hoje. / The aim of this work is to introduce the notion of graphs familiarizing students with a little concept studied in elementary and middle schools. For this, some practical situations were studied and the resolution through graphs. The presentation of the theory of graphs is done using some of the classic problems (The Königsberg bridge problem and The travelling salesman problem) that originated the theory as it is known today.

Leung Chi Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 52-57). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivations --- p.1 / Chapter 1.2 --- Related work --- p.3 / Chapter 1.2.1 --- Fixed-parameter tractability --- p.3 / Chapter 1.2.2 --- Maximum k-vertex cover --- p.4 / Chapter 1.2.3 --- Dominating set --- p.4 / Chapter 1.3 --- Overview of the thesis --- p.5 / Chapter 2 --- Preliminaries --- p.6 / Chapter 2.1 --- Notation and definitions --- p.6 / Chapter 2.1.1 --- Basic definitions --- p.6 / Chapter 2.1.2 --- Partial t-trees --- p.7 / Chapter 2.1.3 --- Cographs --- p.9 / Chapter 2.1.4 --- Chordal graphs and interval graphs --- p.11 / Chapter 2.2 --- Upper bound --- p.12 / Chapter 2.3 --- Extension method --- p.14 / Chapter 3 --- Planar Graphs --- p.17 / Chapter 3.1 --- Trees --- p.17 / Chapter 3.2 --- Partial t-trees --- p.23 / Chapter 3.3 --- Planar graphs --- p.30 / Chapter 4 --- Perfect Graphs --- p.34 / Chapter 4.1 --- Maximum k-vertex cover in cographs --- p.34 / Chapter 4.2 --- Maximum dominating k-set in interval graphs --- p.39 / Chapter 4.3 --- Maximum k-vertex subgraph in chordal graphs --- p.46 / Chapter 4.3.1 --- Maximum k-vertex subgraph in partial t- trees --- p.46 / Chapter 4.3.2 --- Maximum k-vertex subgraph in chordal graphs --- p.47 / Chapter 5 --- Concluding Remarks --- p.49 / Chapter 5.1 --- Summary of results --- p.49 / Chapter 5.2 --- Open problems --- p.50

我們研究與圖的擴張值相關的問題。圖的擴張值反映了一個圖的連通程度。很多不同的應用問題都可以轉換成在圖中找一個不擴張的點集，而這亦跟近似理論中的唯一遊戲猜想(unique games conjecture) 有密切的關係。故此，圖的擴張值在應用和理論上均有研究動機。 / 我們介紹有關圖的擴張值的新發現。這些發現包括設計和分析找不擴張的子集的算法、近似算法的困難結論，以及擴張圖的算法應用： / 我們用更多的特徵值去證明一個Cheeger 不等式的推廣，從而較好地分析譜分割算法(spectral graph partitioning algorithm)。 / 我們分析圖上的隨機漫走來設計一個局部的圖分割算法。此算法跟Cheeger 不等式的保證相合。 / 我們給出圖冪的擴張值的緊下界，並用以得到一個關於找小的不擴張子集的困難結論(hardness result)。 / 我們利用擴張圖來設計一個快而簡單的算法來計算矩陣的秩，新算法顯著地改進了以往的算法。 / We study problems related to graph expansions, which measure how well a graph is connected. Various practical problems can be modelled as finding a small non-expanding subset of vertices, and this is also closely related to the unique games conjecture in the theory of approximation algorithms. Hence our study of graph expansions is both practically and theoretically motivated. / We present our new results about graph expansions. The results include the design and analysis of algorithms for finding non-expanding subsets, hardness of approximation results and algorithmic applications of expanders: / We prove a generalization of Cheeger's inequality using higher eigenvalues, providing a better analysis of the spectral graph partitioning algorithm. / We design a local graph partitioning algorithm using random walks on graph, matching the performance guaranteed by Cheeger's inequality. / We give a tight lower bound for the expansion of graph powers and use it to prove hardness results for small set expansions. / We use expanders to design fast and simple algorithms for computing matrix ranks, significantly improving over previous works. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Kwok, Tsz Chiu. / Thesis (Ph.D.) Chinese University of Hong Kong, 2015. / Includes bibliographical references (leaves 104-109). / Abstracts also in Chinese.