On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

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

Vertex-Magic Total Labeling on G-sun Graphs

Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite sun graphs. We will also show when there is no vertex-magic total labeling for other types of G-sun graphs

Vertex Magic

This paper addresses labeling graphs in such a way that the sum of the vertex labels and incident edge labels are the same for every vertex. Bounds on this so-called magic number are found for cycle graphs. If a graph has an odd number of vertices, algorithms can be found to produce different magic-vertex graphs with the maximum and minimum magic number. Also, every cycle graph with an odd number of vertices can be made into a vertexmagic graph if the odd numbers or even numbers are placed on the vertices. Some interesting problems arise when one begins to look at cycle graphs with an even number of vertices. Bounds for the magic number change, and it becomes harder to make these graphs vertex-magic. We have shown some algorithms for finding vertex-magic cycle graphs with a magic number that lies within the bounds