Maximal nontraceable graphs

A graph G is maximal nontraceable (MNT) (maximal nonhamiltonian (MNH)) if G is not traceable (hamiltonian), i.e. does not contain a hamiltonian path (cycle), but G+xy is traceable (hamiltonian) for all nonadjacent vertices x and y in G. A graph G is hypohamiltonian if G is not hamiltonian, but every vertex deleted subgraph G -u of G is hamiltonian. A graph which is maximal nonhamiltonian and hypohamiltonian is called maximal hypohamiltonian (MHH). Until recently, not much has appeared in the literature about MNT graphs, although there is an extensive literature on MNH graphs. In 1998 Zelinka constructed two classes of MNT graphs and made the conjecture, which he later retracted, that every MNT graph belongs to one of these classes. We show that there are many different types of MNT graphs that cannot be constructed by Zelinka's methods. Although we have not been able to characterize MNT graphs in general, our attempt at characterizing MNT graphs with a specified number of blocks and cut-vertices enabled us to construct infinite families of non-Zelinka MNT graphs which have either two or three blocks. We consider MNT graphs with toughness less than one, obtaining results leading to interesting constructions of MNT graphs, some based on MHH graphs. One result led us to discover a non-Zelinka MNT graph of smallest order, namely of order 8. We also present examples of MNTgraphs with toughness at least one, including an infinite family of 2-connected, claw-free graphs. We find a lower bound for the size of 2-connected MNT graphs of order n. We construct an infinite family of 2-connected cubic MNT graphs of order n, using MHH graphs as building blocks. We thus find the minimum size of 2-connected MNT graphs for infinitely many values of n. We also present a construction, based on MHH graphs, of an infinite family of MNT graphs that are almost cubic. We establish the minimum size of MNT graphs of order n, for all except 26 values of n, and we present a table of MNT graphs of possible smallest size for the excluded 26 values of n.Mathematical SciencesPHD (MATHEMATICS

Local properties of graphs

We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated.Mathematical SciencesD. Phil. (Mathematics

Hamiltonicity of locally hamiltonian and locally traceable graphs

Please read abstract in the article.The University of South Africa and the National Research Foundation of South Africa for their sponsorship of the Salt Rock Workshops of 28 July–10 August 2013 and 20–30 January 2016, which contributed towards results in this paper. The authors thank the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) for financial support, grant number BA2017/268. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS. This material is based upon the third author’s work supported by the National Research Foundation of S.A. under Grant number 81075 and the second author’s work supported by the National Research Foundation of S.A. under Grant number 107668.http://www.elsevier.com/locate/dam2019-02-19hj2018Mathematics and Applied Mathematic

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved

Graphs and subgraphs with bounded degree

"The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modelled by a graph in which vertices represent 'nodes' (stations or processors) while undirected or directed edges stand for 'links' or other types of connections, physical or virtual. A cycle that contains every vertex of a graph is called a hamiltonian cycle and a graph which contains a hamiltonian cycle is called a hamiltonian graph. The problem of the existence of a hamiltonian cycle is closely related to the well known problem of a travelling salesman. These problems are NP-complete and NP-hard, respectively. While some necessary and sufficient conditions are known, to date, no practical characterization of hamiltonian graphs has been found. There are several ways to generalize the notion of a hamiltonian cycle. In this thesis we make original contributions in two of them, namely k-walks and r-trestles." --Abstract.Doctor of Philosoph

On the path sequence of a graph

A subset S of vertices of a graph G = (V;E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by k(G) the minimum cardinality of a k-path vertex cover in G and form a sequence (G) = ( 1(G); 2(G); : : : ; jV j(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in (G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

On the Path Sequence of a Graph

A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by Ψk (G) the minimum cardinality of a k-path vertex cover in G and form a sequence Ψ (G) = (Ψ1 (G), Ψ2 (G), . . . , Ψ|V| (G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in Ψ(G). A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given

A Single Compound Alternative to a Buprenorphine/Naltrexone Combination

Relapse to drug taking is a major factor contributing to the low success rate of opioid addiction treatment programmes. Recently, studies have revealed a buprenorphine/naltrexone combination had successfully increased the treatment retention rate (compared to naltrexone alone) among heroin addicts (with history of cocaine abuse) who had undergone detoxification. However, buprenorphine and naltrexone could not be administered as a single formulation due to their different bioavailability, which could create compliance issues. Therefore, in this project, we aimed to synthesise a series of ligands each having the pharmacological profile of the buprenorphine/naltrexone combination (partial agonist (ORL-1 receptors), antagonist (u- and x-opioid receptors)). Based on the group's previous work, this profile can be achieved within the orvinols series. Compound BU127, a buprenorphine analogue with phenyl substituent (C20) is very close to the desired profile. Therefore, in order to optimize BU127's profile, we designed and synthesised a series of aromatic analogues, including analogues with a small group attached to the aromatic system to increase the ORL-1 receptor efficacy, while retaining the low efficacy / antagonist activity at the u-opioid receptor and antagonist activity at x-opioid receptor. However, [35S]GTPyS screening has shown a sudden increase of x-opioid receptor efficacy with these modifications. The related compound BU10119, having a Cv-methyl, met the desired profile at all targeted receptors in the [35S]GTPyS screen. A few analogues were selected for further evaluation in functional assays in the isolated tissue preparations (rat vas deferens (for the ORL-1 and u-opioid receptors) and mouse vas deferens (for the K-opioid receptor)) to estimate their binding affinity (Ks) and potency (pA2) of the compounds relative to buprenorphine, using Schild analysis and Schild equation. Of the analogues synthesised, only compounds BU127 and BU1 0119 have met the desired profile at the targeted receptors (competitive reversible at the ORL-1 and u-opioid receptors) and having binding affinity at each receptor similar to buprenorphine (ORL-1, ~- and K-opioid receptors). Based on these results, at this point, the optimum features of buprenorphine analogues in order to achieve the targeted profiles are having a small group at Cy and a 6-membered aromatic substituent at C . 20 Without any substituent group attached to the aromatic ring.EThOS - Electronic Theses Online ServiceGBUnited Kingdo