Packings and Coverings of Complete Graphs with a Hole with the 4-Cycle with a Pendant Edge

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w)

Polyhedra, Complexes, Nets and Symmetry

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is "regular" if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well-known to crystallographers, and we identify them explicitly. There also are 6 infinite families of "chiral" apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.Comment: Acta Crystallographica Section A (to appear

Packings and coverings of the complete graph with trees

In this thesis, we define the spectrum problem for packings (coverings) of G to be the problem of finding all graphs H such that a maximum G-packing (minimum G- covering) of the complete graph with the leave (excess) graph H exists. The set of achievable leave (excess) graphs in G-packings (G-coverings) of the complete graph is called the spectrum of leave (excess) graphs for G. Then, we consider this problem for trees with up to five edges. We will prove that for any tree T with up to five edges, if the leave graph in a maximum T-packing of the complete graph Kn has i edges, then the spectrum of leave graphs for T is the set of all simple graphs with i edges. In fact, for these T and i and H any simple graph with i edges, we will construct a maximum T-packing of Kn with the leave graph H. We will also show that for any tree T with k ≤ 5 edges, if the excess graph in a minimum T-covering of the complete graph Kn has i edges, then the spectrum of excess graphs for T is the set of all simple graphs and multigraphs with i edges, except for the case that T is a 5-star, for which the graph formed by four multiple edges is not achievable when n = 12

Multiple minimum coverings of Kn with copies of K4 - e

This paper is the last of a trilogy completely solving the maximum packing and minimum covering problems for the complete graph on n vertices, Kn, with copies of K4 - e, that is, the complete graph on four vertices with one edge missing

PARTIAL COVERING OF A CIRCLE BY EQUAL CIRCLES. PART I: THE MECHANICAL MODELS

How must n equal circles of given radius be placed so that they cover as great a part of the area of the unit circle as possible? To analyse this mathematical problem, mechanical models are introduced. A generalized tensegrity structure is associated with a maximum area configuration of the n circles, whose equilibrium configuration is determined numerically with the method of dynamic relaxation, and the stability of equilibrium is investigated by means of the stiffness matrix of the tensegrity structure. In this Part I, the principles of the models are presented, while an application will be shown in the forthcoming Part II

Epidemic Simulation and Mitigation via Evolutionary Computation

A global pandemic remains a public health event that presents a unique and unpredictable challenge for those making health related decisions and the populations who experience the virus. Though a pandemic also provides the opportunity for researchers and health administrations around the world to mobilize in the fields of epidemiology, computer science, and mathematics to generate epidemic models, vaccines, and vaccination strategies to mitigate unfavourable outcomes. To this end, a generative representation to create personal contact networks, representing the social connections within a population, known as the Local THADS-N generative representation is introduced and expanded upon. This representation uses an evolutionary algorithm and is modified to include new local edge operations improving the performance of the system across several test problems. These problems include an epidemic's duration, spread through a population, and closeness to past epidemic behaviour. The system is further developed to represent sub-communities known as districts, better articulating epidemics spreading within and between neighbourhoods. In addition, the representation is used to simulate four competing vaccination strategies in preparation for iterative vaccine deployment amongst a population, an inevitability when considering the lag inherent to developing vaccines. Finally, the Susceptible-Infected-Removed (SIR) model of infection used by the system is expanded in preparation for adding an asymptomatic state of infection as seen within the COVID-19 pandemic

Chiral polyhedra in ordinary space, II

A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary Euclidean 3-space with finite skew faces and finite skew vertex-figures; they occur in infinite families and are of types {4,6}, {6,4} and {6,6}. Part II completes the enumeration of all discrete chiral polyhedra in 3-space. There exist several families of chiral polyhedra with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I.Comment: 48 page