Graceful labellings of new families of windmill and snake graphs

A function ƒ is a graceful labelling of a graph G = (V,E) with m edges if ƒ is an injection ƒ : V (G) → {0, 1, 2, . . . ,m} such that each edge uv ∈ E is assigned the label |ƒ(u) − ƒ(v)| ∈ {1, 2, . . . ,m}, and no two edge labels are the same. If a graph G has a graceful labelling, we say that G itself is graceful. A variant is a near graceful labelling, which is similar, except the co-domain of f is {0, 1, 2, . . . ,m + 1} and the set of edge labels are either {1, 2, . . . ,m − 1,m} or {1, 2, . . . ,m − 1,m + 1}. In this thesis, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa’s conjecture for a new family of triangular cacti. Further, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with C₃ and C₄ vanes, and infinite families of 3,5-windmills and 3,6-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of t 5-cycles with one vertex in common (Ct₅ ) is graceful if and only if t ≡ 0,3 (mod 4) and near graceful when t ≡ 1, 2 (mod 4). Also, we present a new sufficiency condition to obtain a graceful labelling for every kC₄ₙ snake and use this condition to label every such snake for n = 1, 2, . . . , 6. Then, we extend this result to cyclic snakes where the cycles lengths vary. Also, we obtain new results on the (near) graceful labelling of cyclic snakes based on cycles of lengths n = 6, 10, 14, completely solving the case n = 6

On the Graceful Game

A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

A history of Christchurch home gardening from colonisation to the Queen's visit: gardening culture in a particular society and environment

Garden histories since the mid 1990s have increasingly turned to studies of vernacular gardens as sites of identity formation. More recently, the development of environmental history and specifically urban environmental history has started to show how vernacular gardening in suburban and urban spaces has contributed to changes in urban environments. Relatively little work on home gardening history in this sense has been undertaken in the New Zealand context, while in Australia such work is well underway. This study augments knowledge of home gardening history in New Zealand by focussing on one urban area, Christchurch, known both as the 'Garden City' and as 'one of the most English cities outside of England'. An examination of gardening literature over the period from European colonisation in 1850 to the first visit to the city by a reigning monarch in 1954 highlights changes in gardening tropes rather than particular garden fashions or elements. The four principal tropes of abundance, beauty, protection and sustenance, each supported with a particular kind of ritual-like garden competition, show how gardening discourses related to ideas about the maintenance of the social and cultural order. A more objective measure of attitudes to gardens is gained by examining 1823 property advertisements across the period. Categorised by suburb this analysis shows a level of gardening variation across the city. Following this analysis, case studies of four suburbs in three areas were undertaken. These were based primarily on oral histories and reveal the extent of gardening variation across the city, and the limited but significant effect that gardening discourses had on gardens. This suggests methodological problems with many studies of vernacular gardens, as well as opportunities for further studies. This thesis also demonstrates the value of home gardening histories to urban environmental history, particularly with regard to the former colonies of the British Empire

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

‘Travelling Without Moving’ and ‘Transformation, Identity and Liminality in Travel Memoir’

This thesis comprises a creative work and an exegesis. The creative work, Travelling Without Moving, is an autobiographically-inspired, fictional, discontinuous travel narrative that explores why and how we navigate our inner and outer journeys, and correlates the suffering of the world that surrounds us with the uncharted elements—confusion, passion, evil—of our inner lives. The exegesis considers how travel memoirs question our liminal encounters with the ‘other’ and the affective transformation of our identity