Universiẗat, Diss., 1999--Dortmund. / Dateiformat: PDF.

This paper introduces the hypergraph transversal problem along with thefollowing iterative solutions: naive, branch and bound, and dynamic exponentialtime (NC-D). Odometers are introduced along with the functions that manipulatethem. The traditional definitions of hyperedge, hypergraph, etc., are redefined interms of odometers and lists. All algorithms and functions necessary to implementthe solution are presented along with techniques to validate and test the results.Lastly, parallelization advanced applications, and future research directions areexamined.

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超級混合圖是一個 H = (X,C,D) 的表示法，其中X是代表點集合，而C和D是X的部分子集合，稱為邊。一個嚴格k種顏色可著色法指的是由X的點集對應到{1,2,…,k}的一種關係，其中C代表每一個C邊至少有兩個點同色，而D代表每一個D邊至少有兩個點不同色。C和D都有可能是空集合。假如超過(少於)k並沒有可著色的方法數，則k稱為最大著色數(最小著色數)。而H的每個邊都恰好有r個點則稱為r均勻超級混合圖。 對於r均勻C超級混合圖，如果限定了最大著色數大於等於k的話，則將會改變最大著色數的邊數。如果要找出滿足此條件的最大著色數的最大的邊數，我們主要區分成三種不同的情形來討論，分別是r比k大、r比k小和r = k。 / A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, and each of C,D is a list of subsets of X. A strict k-coloring is a onto mapping from X to {1,2, . . . , k} such that each C ∈ C contains two vertices have a common value and each D ∈ D has two vertices have distinct values. Each of C,D may be empty. The maximum(minimum) number of colors over all strict k-colorings is called the upper(lower) chromatic number of H and is denoted by χ^¯(H)(χ(H)). If a hypergraph H has no multiple edges and all its edges are of size r, then H is called an r-uniform hypergraph. We want to find the maximum number of edges for r-uniform C-hypergraph of order n with the condition χ^¯(H) ≥ k, where k is fixed. We will solve this problem according to three different cases, r < k, r = k and r > k.

<p>Brain connectome network visualization could help the neurologists inspect the brain structure easily and quickly. In the thesis, the model of the brain connectome network is visualized in both three dimensions (3D) environment and two dimensions (2D) environment. One is named “Brain Explorer for Connectomic Analysis” (BECA) developed by the previous research already. It could present the 3D model of brain structure with region of interests (ROIs) in different colors [5]. The other is mainly for the information visualization of brain connectome in 2D. It adopts the force-directed layout to visualize the network. However, the brain network visualization could not bring the user intuitively ideas about brain structure. Sometimes, with the increasing scales of ROIs (nodes), the visualization would bring more visual clutter for readers [3]. So, brain connectome network properties visualization becomes a useful complement to brain network visualization. For a better understanding of the effect of Alzheimer’s disease on the brain nerves, the thesis introduces several methods about the brain graph properties visualization. There are the five selected graph properties discussed in the thesis. The degree and closeness are node properties. The shortest path, maximum flow, and clique are edge properties. Except for clique, the other properties are visualized in both 3D and 2D. The clique is visualized only in 2D. For the clique, a new hypergraph visualization method is proposed with three different algorithms. Instead of using an extra node to present a clique, the thesis uses a “belt” to connect all nodes within the same clique. The methods of node connections are based on the traveling salesman problem (TSP) and Law of cosines. In addition, the thesis also applies the result of the clique to adjust the force-directed layout of brain graph in 2D to dramatically eliminate the visual clutter. Therefore, with the support of the graph properties visualization, the brain connectome network visualization tools become more flexible.</p>

abstract: Extremal graph theory results often provide minimum degree conditions which guarantee a copy of one graph exists within another. A perfect $F$-tiling of a graph $G$ is a collection $\mathcal{F}$ of subgraphs of $G$ such that every element of $\mathcal{F}$ is isomorphic to $F$ and such that every vertex in $G$ is in exactly one element of $\mathcal{F}$. Let $C^{3}_{t}$ denote the loose cycle on $t = 2s$ vertices, the $3$-uniform hypergraph obtained by replacing the edges $e = \{u, v\}$ of a graph cycle $C$ on $s$ vertices with edge triples $\{u, x_e, v\}$, where $x_e$ is uniquely assigned to $e$. This dissertation proves for even $t \geq 6$, that any sufficiently large $3$-uniform hypergraph $H$ on $n \in t \mathbb{Z}$ vertices with minimum $1$-degree $\delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) + 1$, where $c(t,n) \in \{0, 1, 3\}$, contains a perfect $C^{3}_{t}$-tiling. The result is tight, generalizing previous results on $C^3_4$ by Han and Zhao. For an edge colored graph $G$, let the minimum color degree $\delta^c(G)$ be the minimum number of distinctly colored edges incident to a vertex. Call $G$ rainbow if every edge has a unique color. For $\ell \geq 5$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + 1}{2}$ contains a rainbow cycle on $\ell$ vertices. The result is tight for odd $\ell$ and extends previous results for $\ell = 3$. In addition, for even $\ell \geq 4$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + c(\ell)}{3}$, where $c(\ell) \in \{5, 7\}$, contains a rainbow cycle on $\ell$ vertices. The result is tight when $6 \nmid \ell$. As a related result, this dissertation proves for all $\ell \geq 4$, that any sufficiently large oriented graph $D$ on $n$ vertices with $\delta^+(D) \geq \frac{n + 1}{3}$ contains a directed cycle on $\ell$ vertices. This partially generalizes a result by Kelly, K\"uhn, and Osthus that uses minimum semidegree rather than minimum out degree. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

We consider tiling problems for graphs and hypergraphs. For two graphs and , an -tiling of is a subgraph of consisting of only vertex disjoint copies of . By using the absorbing method, we give a short proof that in a balanced tripartite graph , if every vertex is adjacent to of the vertices in each of the other vertex partitions, then has a -tiling. Previously, Magyar and Martin [11] proved the same result (without ) by using the Regularity Lemma. In a 3-uniform hypergraph , let denote the minimum number of edges that contain for all pairs of vertices. We show that if , there exists a -tiling that misses at most vertices of . On the other hand, we show that there exist hypergraphs such that and does not have a perfect -tiling. These extend the results of Pikhurko [12] on -tilings.

In this thesis are analysed correctness for verification ability of UML activity diagrams by using hypergaphs. There suggested the transformation from UML activity diagram to hypergraph and verification algorithm of hypergraph, which can be used for expanding the capabilities of CASE tools for workflow and e-business modelling.

A generalization of the circular chromatic number to hypergraphs is devel-oped. Circular colourings of graphs and hypergraphs are first discussed and it is shown that the circular chromatic number of a graph is the same regard-less of whether the hypergraph or graph definition is used. After presenting a few basic results, some examples of circular chromatic numbers of various families of hypergraphs are given. Subsequently, the concepts of the star chromatic number and the arc chromatic number are introduced. Specif¬ically, both numbers are shown to be equivalent to the circular chromatic number. Finally the relationship between the imbalance of a hypergraph and the circular chromatic number is explored and a classical result of Minty is deduced.