k-ordered Graphs & Out-arc Pancyclicity on Digraphs

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Preface Graph theory as a very popular area of discrete mathematics has rapidly been developed over the last couple of decades. Numerous theoretical results and countless applications to practical problems have been discovered. The concepts of k-ordered graphs and out-arc pancyclicity are two recent topics in graph theory, which are investigated in this thesis. Hamiltonian graphs and various Hamiltonian-related concepts such as traceable-, Hamiltonian-connected-, pancyclic-, panconnected-, and cycle extendable graphs have been studied extensively. Recently, Ng and Schulz [62] introduced a new strong Hamilto-nian property: k-ordered Hamiltonian. For a positive integer k, a graph G is k-ordered if for every ordered set of k vertices, there is a cycle that encounter the vertices of the set in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonian. Just as many articles showed for other Hamiltonian-related properties, the condition that implies a graph to be Hamiltonian is a candidate to imply a graph to be k-ordered Hamiltonian. There has been a series of results involving degree conditions, generalized degree conditions, neighbourhood conditions and forbidden subgraph conditions that imply k-ordered or k-ordered Hamiltonian (see [23]). In Part I, some new results on k-ordered graphs are presented. Chapter 1 gives a general introduction to the terminology, notation and basic concepts of k-ordered (Hamiltonian) graphs. Related useful results on Hamiltonicity and k-ordered Hamiltonicity are also recalled. In Chapter 2, connectivity properties of k-ordered graphs are investigated. Section 2.1 deals with the minimum connectivity forced by a k-ordered graph. To this aim, we introduce a new kind of connectivity: k-ordered connectivity. The concept of k-ordered graphs is related to other " connectivity " concepts, such as, linkage. In Section 2.2, relationships between connectivity, linkage and orderedness are described. Chen et al. [18] showed the absorptivity of k-linked graphs. In section 2.3, we show a similar absorptivity on k-ordered graphs. It is well known that the Hamiltonicity problem is NP-complete. There are many conditions that imply a graph to be Hamiltonian. One of the classical theorems of this nature is due to Ore [64], who proved that a graph is Hamiltonian if the degree sum of any two nonadjacent vertices is at least n. In Chapter 3, we shall not consider " any pair of nonadjacent vertices " , but only " any pair of distance 2 …