On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

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

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

Nested locally Hamiltonian graphs and the Oberly-Sumner conjecture

Please read abstract in the article.The DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) and the National Research Foundation of S.A.https://www.dmgt.uz.zgora.plam2023Mathematics 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