Nowadays, the bevel gears are widely applied in the industry for the intersected-axial transmission system. But the applications of the bevel gears are mostly limited to the usage of involute bevel gears. In this thesis, the bevel gear with circular-arc tooth profiles is derived by using general theorem of conjugate surfaces, coordinate transformation, constrained meshing equation, and spherical trigonometry. According to the bevel gears with circular-arc tooth profile derived above, the analyses and discussions of the interference are proposed. The interference situation is detected by applying the phase lead-lag concept while circular-arc curve is moving on the spherical cross-section. Furthermore, the ideal conditions to avoid occurrence of interference are proposed. Design charts for the maximum values of tooth profile angle are also constructed as a reference for designers. The 3D solid models of the bevel gear with circular-arc tooth profiles are constructed by using the computer software (Pro/E). Finally, the transmission ability is verified through the computer animation using CAE software (Visual Nastran). It is believed that the mathematical models and design method developed in the thesis will provide a useful foundation for the further studies.

The bevel gears have been widely used for the intersected-axial transmission system for a long time. But mostly them are limited in the category of involute tooth profiles. It is believed that bevel gears with circular-arc tooth profile, similar to the Wildhaber-Novikov circular-arc helical gears, will improve the load capacity of the gear set. The bevel gears with circular-arc tooth profiles was firstly proposed by Kuo and Tsai in 2001. Although these new type of bevel gears can increase the load capacity of transmission, the expensive manufacturing process is still the problem. In this paper, the design parameters of bevel gear with circular-arc profiles developed by Kuo and Tsai are modified. Bevel gear set with spiral point contact path is developed. This improvement makes it possible to manufacturing the newly developed bevel gears in just a simple milling or/and grinding process with circular cutting edges. The manufacturing process is then cost down quit a lot. A method for checking the gear interference is also proposed. Finally, the 3D solid models of the bevel gear with circular-arc tooth profiles as well as the grinding wheel are constructed by using the computer software ¡§Pro/E¡¨. It is believed that the mathematical models and the design method developed in the thesis will provide a useful foundation for the further studies.

Metal deposition is an additive layered manufacturing process that deposits molten metal droplets on a substrate and by repeating this process layer by layer, a complex shaped 3D geometry can be manufactured. In this thesis, the metal deposition process is performed by a robot with a wire feeder tool and a laser as energy source to melt the metal wire. The robot programming for robotized metal deposition process can be completely automated by computer aided robotics software. University West is currently developing an add-in application in a computer aided robotics software, Process Simulate, that is capable of programming the robotized metal deposition process. The first goal of this thesis was to verify the up to now developed software and the process from CAD drawing down to robot code. Another goal was to find and implement an algorithm that will reduce the number of locations on a circular arc to three locations. The algorithm to minimize the locations must be capable of changing all the different curvature paths to linear and circular arc motions which are easy to translate to robot code. The user should be able to decide the fitting precision of the approximated motion path to the original path. A real robot cell setup is modelled in Process Simulate. This lets Process Simulate generate the correct robot code for that specific cell.  Since each robot cell has its own unique setup, a custom script will be developed that changes the universal robot code, that Process Simulate generates, to the custom robot code required in this specific robot cell. The software is improved and tested from CAD drawing down to robot code but still needs to be debugged more and needs implementation of some non-existing features.

abstract: A central concept of combinatorics is partitioning structures with given constraints. Partitions of on-line posets and on-line graphs, which are dynamic versions of the more familiar static structures posets and graphs, are examined. In the on-line setting, vertices are continually added to a poset or graph while a chain partition or coloring (respectively) is maintained. %The optima of the static cases cannot be achieved in the on-line setting. Both upper and lower bounds for the optimum of the number of chains needed to partition a width $w$ on-line poset exist. Kierstead's upper bound of $\frac{5^w-1}{4}$ was improved to $w^{14 \lg w}$ by Bosek and Krawczyk. This is improved to $w^{3+6.5 \lg w}$ by employing the First-Fit algorithm on a family of restricted posets (expanding on the work of Bosek and Krawczyk) . Namely, the family of ladder-free posets where the $m$-ladder is the transitive closure of the union of two incomparable chains $x_1\le\dots\le x_m$, $y_1\le\dots\le y_m$ and the set of comparabilities $\{x_1\le y_1,\dots, x_m\le y_m\}$. No upper bound on the number of colors needed to color a general on-line graph exists. To lay this fact plain, the performance of on-line coloring of trees is shown to be particularly problematic. There are trees that require $n$ colors to color on-line for any positive integer $n$. Furthermore, there are trees that usually require many colors to color on-line even if they are presented without any particular strategy. For restricted families of graphs, upper and lower bounds for the optimum number of colors needed to maintain an on-line coloring exist. In particular, circular arc graphs can be colored on-line using less than 8 times the optimum number from the static case. This follows from the work of Pemmaraju, Raman, and Varadarajan in on-line coloring of interval graphs. / Dissertation/Thesis / Ph.D. Mathematics 2012

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Conjectured in 1943, Hadwiger’s conjecture is one of the most challenging open problems in graph theory. Hadwiger’s conjecture states that if the chromatic number of a graph G is k, then G has a clique minor of size at least k. In this thesis, we give motivation for attempting Hadwiger’s conjecture for circular arc graphs and also prove the conjecture for proper circular arc graphs. Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are a subclass of circular arc graphs that have a circular arc representation where no arc is completely contained in any other arc. It is interesting to study Hadwiger’s conjecture for circular arc graphs as their clique minor cannot exceed beyond a constant factor of its chromatic number as We show in this thesis. Our main contribution is the proof of Hadwiger’s conjecture for proper circular arc graphs. Also, in this thesis we give an analysis and some basic results on Hadwiger’s conjecture for circular arc graphs in general.

Beim Graphisomorphieproblem geht es um die Frage, ob zwei Graphen bis auf Knotenumbenennungen die gleiche Struktur haben. Es ist eines der wenigen verbleibenden natürlichen Probleme, für die weder ein Polynomialzeitalgorithmus noch NP-Härte bekannt ist. Aus dieser Situation ist ein Forschungszweig erwachsen, der effiziente Isomorphiealgorithmen für eingeschränkte Graphklassen entwickelt. Der Hauptbeitrag dieser Arbeit besteht in Logspace-Algorithmen, die das Isomorphieproblem für k-Bäume, Intervallgraphen, sowie Helly- und Proper-Kreisbogengraphen lösen. Dies verbessert zuvor bekannte parallele Algorithmen und führt zu einer vollständigen Klassifikation der Komplexität dieser Probleme, da für sie auch Logspace-Härte nachgewiesen wird. Tatsächlich leisten die vorgestellten Algorithmen mehr: Im Fall der k-Bäume berechnet der Algorithmus kanonische Knotenbenennungen mit O(k log n) Platz. Eine alternative Implementation des Algorithmus kommt mit O((k+1)!n) Zeit aus – hierbei ist n die Anzahl der Knoten – und ist damit der schnellste bekannte FPT-Algorithmus für Isomorphie von k-Bäumen. Die Algorithmen für Intervall- und Kreisbogengraphen berechnen kanonische Repräsentationen – das heißt, sie weisen jedem Knoten ein Intervall (beziehungsweise einen Kreisbogen) zu, sodass diese sich genau dann schneiden, wenn die zugehörigen Knoten benachbart sind, und isomorphe Eingabegraphen das gleiche Intervallmodell (beziehungsweise Kreisbogenmodell) erhalten. Außerdem werden auch Logspace-Algorithmen angegeben, die Intervallrepräsentationen mit zusätzlichen Eigenschaften berechnen – oder erkennen, dass dies nicht möglich ist: Für die resultierenden Intervallmodelle kann gefordert werden, dass sie proper sind (also kein Intervall ein anderes enthält), dass sie unit sind (also alle Intervalle die gleiche Länge haben) oder dass die Längen der paarweisen Schnitte (und optional der einzelnen Intervalle) vorgegebenen Werten entsprechen. / The graph isomorphism problem deals with the question if two graphs have the same structure up to renaming their vertices. It is one of the few remaining natural problems for which neither a polynomial-time algorithm nor NP-hardness is known. This situation has led to a branch of research that develops efficient algorithms for special cases of the graph isomorphism problem, where the input graphs are required to be from restricted graph classes. The main contribution of this thesis comprises of logspace algorithms that solve the isomorphism problem for k-trees, interval graphs, Helly circular-arc graphs and proper circular-arc graphs. This improves previously known parallel algorithms and leads to a complete classification of the complexity of these problems, as they are also shown to be hard for logspace. In fact, these algorithms achieve more: In the case of k-trees, the algorithm computes canonical labelings in space O(k log n). An alternative implementation runs in time O((k+1)!n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism. The algorithms for interval and circular-arc graphs actually compute canonical representations, i.e., each vertex is assigned an interval (or arc) such that these intersect each other if and only if the corresponding vertices are adjacent, and isomorphic input graphs receive the same interval (or arc) model. This thesis also presents logspace algorithms that compute interval representations with additional properties, or detect that this is not possible: The resulting interval models can be required to be proper (no interval contains another), unit (all intervals have the same length), or to satisfy prescribed lengths for pairwise intersections (and possibly prescribed lengths of intervals).

The orientation completion problem for a hereditary class C of oriented graphs asks whether a given partially oriented graph can be completed to a graph belonging to C. This problem was introduced recently and is a generalization of several existing problems, including the recognition problem for certain classes of graphs and the representation extension problem for proper interval graphs. A local tournament is an oriented graph in which the in-neighbourhood as well as the out-neighbourhood of each vertex induces a tournament. Local tournaments are a well-studied class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tournaments. The orientation completion problems for the class of local tournaments and the class of acyclic local tournaments have been shown to be polynomial-time solvable. In this thesis, we characterize the partially oriented graphs that can be completed to local tournaments by finding a complete list of obstructions. These are in a sense the minimal partially oriented graphs that cannot be completed to local tournaments. We also determine the minimal partially oriented graphs that cannot be completed to acyclic local tournaments. / Graduate

This thesis mainly focuses on algorithmic and combinatorial questions related to some geometric problems on graphs. In the last part of this thesis, a graph coloring problem is also discussed. Boxicity and Cubicity: These are graph parameters dealing with geomet-ric representations of graphs in higher dimensions. Both these parameters are known to be NP-Hard to compute in general and are even hard to approximate within an O(n1− ) factor for any > 0, under standard complexity theoretic assumptions. We studied algorithmic questions for these problems, for certain graph classes, to yield efficient algorithms or approximations. Our results include a polynomial time constant factor approximation algorithm for computing the cubicity of trees and a polynomial time constant (≤ 2.5) factor approximation algorithm for computing the boxicity of circular arc graphs. As far as we know, there were no constant factor approximation algorithms known previously, for computing boxicity or cubicity of any well known graph class for which the respective parameter value is unbounded. We also obtained parameterized approximation algorithms for boxicity with various edit distance parameters. An o(n) factor approximation algorithm for computing the boxicity and cubicity of general graphs also evolved as an interesting corollary of one of these parameterized algorithms. This seems to be the first sub-linear factor approximation algorithm known for computing the boxicity and cubicity of general graphs. Planar grid-drawings of outerplanar graphs: A graph is outerplanar, if it has a planar embedding with all its vertices lying on the outer face. We give an efficient algorithm to 2-vertex-connect any connected outerplanar graph G by adding more edges to it, in order to obtain a supergraph of G such that the resultant graph is still outerplanar and its pathwidth is within a constant times the pathwidth of G. This algorithm leads to a constant factor approximation algorithm for computing minimum height planar straight line grid-drawings of outerplanar graphs, extending the existing algorithm known for 2-vertex connected outerplanar graphs. n−1 3 Maximum matchings in triangle distance Delaunay graphs: Delau-nay graphs of point sets are well studied in Computational Geometry. Instead of the Euclidean metric, if the Delaunay graph is defined with respect to the convex distance function defined by an equilateral triangle, it is called a Trian-gle Distance Delaunay graph. TD-Delaunay graphs are known to be equivalent to geometric spanners called half-Θ6 graphs. It is known that classical Delaunay graphs of point sets always contain a near perfect matching, for non-degenerate point sets. We show that Triangle Distance Delaunay graphs of a set of n points in general position will always l m contain a matching of size and this bound is tight. We also show that Θ6 graphs, a class of supergraphs of half-Θ6 graphs, can have at most 5n − 11 edges, for point sets in general position. Heterochromatic Paths in Edge Colored Graphs: Conditions on the coloring to guarantee the existence of long heterochromatic paths in edge col-ored graphs is a well explored problem in literature. The objective here is to obtain a good lower bound for λ(G) - the length of a maximum heterochro-matic path in an edge-colored graph G, in terms of ϑ(G) - the minimum color degree of G under the given coloring. There are graph families for which λ(G) = ϑ(G) − 1 under certain colorings, and it is conjectured that ϑ(G) − 1 is a tight lower bound for λ(G). We show that if G has girth is at least 4 log2(ϑ(G))+2, then λ(G) ≥ ϑ(G)− 2. It is also proved that a weaker requirement that G just does not contain four-cycles is enough to guarantee that λ(G) is at least ϑ(G) −o(ϑ(G)). Other special cases considered include lower bounds for λ(G) in edge colored bipartite graphs, triangle-free graphs and graphs without heterochromatic triangles.