Graph Theory

Developments in graph colouring theory were motivated by the four-colour problem and Heawood's theorem. Both of these were originally formulated as map-colouring problems that can be expressed as colouring graphs embedded on surfaces. This chapter gives an overview of the abundance of results concerning the chromatic number of graphs that are embedded on surfaces . Introduction In 1852 Francis Guthrie asked whether the regions of every planar map can be coloured with four colours in such a way that no two regions of the map with common boundary receive the same colour. In effect, by duality Guthrie was asking whether every planar graph is 4-colourable. This easily stated problem became known as the famous four-colour problem. Attempts to solve it led to many important results in graph theory, but the problem itself remained unsolved for more than a century. It was finally answered in the positive by Appel and Haken [9], [10], [12]. More about its proof comes later in this chapter. For more than a century, the four-colour problem was one of the driving forces that led to developments in graph theory, and specifically graph colouring theory. Its generalization – the Heawood problem – was the main motivation for developments in topological graph theory (see [51] or [49]). Both of these were originally formulated as map-colouring problems, asking about colouring the faces of an embedded graph so that adjacent faces receive different colours. Every map-colouring problem can be expressed as a graph-colouring problem by considering the dual graph of the map. Its vertices correspond to the faces of the map, and two vertices are adjacent if the corresponding faces are adjacent on the surface. In this chapter we discuss the colouring of graphs embedded on surfaces. We start by describing some of the ideas in the proof of the four-colour theorem and give a proof of the list-colouring version of the five-colour theorem that is due to Thomassen.

The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

Graph theoretical studies have been designed to investigate network topologies during life. Network science and graph theory methods may contribute to a better understanding of brain function, both normal and abnormal, throughout developmental stages. The degree to which childhood epilepsies exert a significant effect on brain network organisation and cognition remains unclear. The hypothesis suggests that the formation of abnormal networks associated with epileptogenesis early in life causes a disruption in normal brain network development and cognition, reflecting abnormalities in later life. Neurological diseases with onset during critical stages of brain maturation, including childhood epilepsy, may threaten this orderly neurodevelopmental process. According to the hypothesis that the formation of abnormal networks associated with epileptogenesis in early life causes a disruption in normal brain network development, it is then mandatory to perform a proper examination of children with new-onset epilepsy early in the disease course and a deep study of their brain network organisation over time. In regards, graph theoretical analysis could add more information. In order to facilitate further development of graph theory in childhood, we performed a systematic review to describe its application in functional dynamic connectivity using electroencephalographic (EEG) analysis, focussing on paediatric epilepsy.

A hypergraph is a generalization of a graph in which edges—called hyperedges—can connect any number of vertices, not just two. A superhypergraph further extends this framework by introducing recursive powerset constructions, enabling the representation of hierarchical relationships among hyperedges themselves. A topological graph is a graph embedded in the plane, where each vertex is represented by a distinct point and each edge is drawn as a continuous curve connecting its endpoints. A topological hypergraph extends this idea by representing hyperedges as closed curves that enclose sets of vertices on the plane [13]. In this paper, we introduce the concept of a Topological n-SuperHypergraph, which unifies the structural hierarchy of n-superhypergraphs with the geometric intuition of topological hypergraphs. We anticipate that this new formulation will contribute to future developments in graph theory, topology, and network theory.

A jump to the bell number for hereditary graph properties

Minimal reducible bounds for induced-hereditary properties

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

Let p be a hereditary graph property. The p‐chromatic number of a graph is the minimal number of classes in a vertex partition wherein each class spans a subgraph with property p. For the property p of edgeless graphs the p‐chromatic number is just the usual chromatic number, whose value is known to be (1/2 + o(1))n/log2 n for almost every graph of order n. We show that we may associate with every nontrivial hereditary property p an explicitly defined natural number r = r(p), and that the p‐chromatic number is then (l/2r + o(1))n/log2 n for almost every graph of order n. © 1995 John Wiley & Sons, Inc.

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let P1, . . . ,Pn be additive hereditary graph properties. A graph G has property (P1 ◦ · · · ◦Pn) if there is a partition (V1, . . . , Vn) of V (G) into n sets such that, for all i, the induced subgraph G[Vi] is in Pi. A property P is reducible if there are properties Q, R such that P = Q ◦R; otherwise it is irreducible. Mihok, Semanǐsin and Vasky [J. Graph Theory 33 (2000), 44–53] gave a factorisation for any additive hereditary property P into a given number dc(P) of irreducible additive hereditary factors. Mihok [Discuss. Math. Graph Theory 20 (2000), 143–153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and ∗The first author’s studies in Canada are fully funded by the Canadian government through a Canadian Commonwealth Scholarship. The second author’s research is financially supported by NSERC. The results presented here form part of the first author’s Ph.D. thesis, that he is writing under the supervision of the second author.

Development of process planning methods is an important research area. Product development planning takes into consideration lead-times of tasks and activities, bill of materials, quality plans, and risk analysis. Production process planning can be supported by graph theory. Methods such as CPM, PERT, GERT are useful in product development planning. In this article, a methodology of risk identification and assessment is combined with the GERT method.

The workshop Algorithmic Graph Theory , organized by Artur Czumaj (New Jersey), Friedhelm Meyer auf der Heide (Paderborn), Klaus Jansen (Kiel) and Ingo Schiermeyer (Freiberg) was held February 12th–February 18th, 2006. This meeting was attended by 46 participants from a wide range of geographic regions, many of them young researchers. In the morning sessions survey talks providing an overview of recent developments in Algorithmic Graph Theory were presented. Geometry and Graphs Graph Algorithms Game Theory and Graphs Graph Structures There were 10 shorter talks and two special sessions on “Graph Algorithms” (organized by Christian Sohler) and “Graph Colourin” (organized by Ingo Schiermeyer). The contributions showed progress in the field provided in recent years. Furthermore, several open problems and conjectures were presented, some of them still far from being resolved. Beyond the program there was plenty of time for the participants to use the stimulating atmosphere of the Oberwolfach Institute for fruitful discussions.

Conceptual configuration synthesis of line-foldable type quadrangular prismatic deployable unit based on graph theory

Graph theory is an integral part of a discrete structures course required in computer science programs. Unfortunately, in the short time such courses devote to graph theory, students often do not appreciate and/or recognize how graph theory notions and theorems become problem modeling and problem solving tools. Graph Works is a visualization tool to show what properties of graph mean in terms of a particular graph and what properties and substructures fundamental theorems guarantee to exist in a graph. The tool is designed to help users understand and develop algorithms that use graph properties needed for problem solving or find and use particular subgraphs as part of the solution process.

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems.

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris–Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors.