A SURVEY OF DISTANCE MAGIC GRAPHS

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x) where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph. In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs. In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants. In Chapter 3, we study graph labelings that retain the same labels as distance magic labelings, but alter the definition in some other way. These labelings include balanced distance magic labelings, closed distance magic labelings, D-distance magic labelings, and distance antimagic labelings. In Chapter 4, we examine results on neighborhood magic labelings, group distance magic labelings, and group distance antimagic labelings. These graph labelings change the label set, but are otherwise similar to distance magic graphs. In Chapter 5, we examine some applications of distance magic and distance antimagic labeling to the fair scheduling of tournaments. In Chapter 6, we conclude with some open problems

Distance magic-type and distance antimagic-type labelings of graphs

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized for directed graphs; the weight will take into consideration the direction of the arcs. In this manuscript, we provide new results for d-handicap labeling, a distance antimagic-type labeling, and introduce a new distance magic-type labeling called orientable Gamma-distance magic labeling. A d-handicap distance antimagic labeling (or just d-handicap labeling for short) of a graph G=(V,E) of order n is a bijection f from V to {1,2,...,n} with induced weight function w(x_{i})=\underset{x_{j}\in N(x_{i})}{\sum}f(x_{j}) \] such that f(x_{i})=i and the sequence of weights w(x_{1}),w(x_{2}),...,w(x_{n}) forms an arithmetic sequence with constant difference d at least 1. If a graph G admits a d-handicap labeling, we say G is a d-handicap graph. A d-handicap incomplete tournament, H(n,k,d) is an incomplete tournament of n teams ranked with the first n natural numbers such that each team plays exactly k games and the strength of schedule of the ith ranked team is d more than the i+1st ranked team. That is, strength of schedule increases arithmetically with strength of team. Constructing an H(n,k,d) is equivalent to finding a d-handicap labeling of a k-regular graph of order n. In Chapter 2 we provide general constructions for every d at least 1 for large classes of both n and k, providing breadth and depth to the catalog of known H(n,k,d)\u27s. In Chapters 3 - 6, we introduce a new type of labeling called orientable Gamma-distance magic labeling. Let Gamma be an abelian group of order n. If for a graph G=(V,E) of order n there exists an orientation of G and a companion bijection f from V to Gamma with the property that there is an element mu in Gamma (called the magic constant) such that \[ w(x)=\sum_{y\in N_{G}^{+}(x)}\overrightarrow{f}(y)-\sum_{y\in N_{G}^{-}(x)}\overrightarrow{f}(y)=\mu for every x in V where w(x) is the weight of vertex x, we say that G is orientable Gamma-distance magic}. In addition to introducing the concept, we provide numerous results on orientable Z_n distance magic graphs, where Z_n is the cyclic group of order n. In Chapter 7, we summarize the results of this dissertation and provide suggestions for future work