published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

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The distance-two constrained labeling of graphs arises in the context of frequency assignment problem (FAP) in mobile and wireless networks. The frequency assignment problem is the problem of assigning frequencies to the stations of a network, so that interference between nearby stations is avoided or minimized while the frequency reusability is exploited. It was first formulated as a graph coloring problem by Hale, who introduced the notion of the T-coloring of a graph, and that attracts a lot of interest in graph coloring. In 1988, Roberts proposed a variation of the channel assignment problem in which “close transmitters must receive different channels and “very close transmitters must receive channels at least two apart. Motivated by this variation, Griggs and Yeh first proposed and studied the L(2, 1)-labeling of a simple graph with a condition at distance two. Because of practical and theoretical applications, the interest for distance-two constrained labeling of graphs is increasing. Since then, many aspects of the problem and related problems remain to be further explored. In this thesis, we first give an upper bound of the L(2, 1)-labeling number, or simply λ number, for a special class of graphs, the n-cubes Qn, where n = 2k k 1. Chang et al. [3] considered a generalization of L(2, 1)-labeling, namely, L(d, 1)- labeling of graphs. We study the L(1, 1)-labeling number of Qn. A lower bound onλ1(Qn) is provided and λ1(Q2k1) is determined. As a related problem, the L(2, 1)-choosability of graphs is studied. Vizing [17] and Erdos et al. [18] generalized the graph coloring problem and introduced the list coloring problem independently more than three decades ago. We shall consider a new variation of the L(2, 1)-labeling problem, the list-L(2, 1)-labeling problem. We determine the L(2, 1)-choice numbers for paths and cycles. We also study the L(2, 1)- choosability for some special graphs such as the Cartesian product graphs and the generalized Petersen graphs. We provide upper bounds of the L(2, 1)-choice numbers for the Cartesian product of a path and a spider, also for the generalized Petersen graphs. Keywords: distance-two labeling, λ-number, L(2, 1)-labeling, L(d, 1)-labeling, list-L(2, 1)-labeling, choosability, L(2, 1)-choice number, path, cycle, n-cube, spider, Cartesian product graph, generalized Petersen graph.

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Research Doctorate - Doctor of Philosophy (PhD) / In this thesis, a number of new methods for constructing vertex-magic total-labelings of graphs are presented. These represent an advance on existing methods since they are general constructions rather than ad hoc constructions for specific families of graphs. Broadly, five new kinds of construction methods are presented. Firstly, we present a class of methods characterized by adding 2- or 4-factors to a labeled graph, reassigning vertex labels to the edges of these factors and then adding new vertex labels to create a VMTL of the new graph. The major result is a unified method for constructing VMTL of large families of regular graphs, providing strong evidence for MacDougall's conjecture that, apart from a few minor exceptions, all regular graphs possess vertex-magic total-labelings. Secondly, we present methods for obtaining a labeling of a union of two graphs, one of which possesses a strong labeling, and then building on this labeling to create a labeling of an irregular graph. These methods as well as results in the Appendices provide strong evidence against an early conjecture regarding labelings and vertex degrees. Thirdly, constructions are presented for a new kind of magic square, containing some zeroes, which can be used to build labelings of graphs from labeled spanning subgraphs. Next, constructions are presented for a new kind of anti-magic square, containing some zeroes, which is equivalent to a strong labeling of certain kinds of bipartite graphs which can in turn be built upon to produce labelings of graphs with more edges. Finally, we present a method of mutating a graph labeling by reassigning edges in a way that preserves the magic constant to obtain a labeling of a different graph. This method provides a prolific source of new labelings.