A generalization of Ore's Theorem involving neighborhood unions

AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2

On the size of maximally non-hamiltonian digraphs

A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open

On factors of 4-connected claw-free graphs

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is Hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e. graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths. \u

Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Secure and robust multi-constrained QoS aware routing algorithm for VANETs

Secure QoS routing algorithms are a fundamental part of wireless networks that aim to provide services with QoS and security guarantees. In Vehicular Ad hoc Networks (VANETs), vehicles perform routing functions, and at the same time act as end-systems thus routing control messages are transmitted unprotected over wireless channels. The QoS of the entire network could be degraded by an attack on the routing process, and manipulation of the routing control messages. In this paper, we propose a novel secure and reliable multi-constrained QoS aware routing algorithm for VANETs. We employ the Ant Colony Optimisation (ACO) technique to compute feasible routes in VANETs subject to multiple QoS constraints determined by the data traffic type. Moreover, we extend the VANET-oriented Evolving Graph (VoEG) model to perform plausibility checks on the exchanged routing control messages among vehicles. Simulation results show that the QoS can be guaranteed while applying security mechanisms to ensure a reliable and robust routing service