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.
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Daisy Cunningham, Vertex Magic, Furman University Electronic Journal of Undergraduate Mathematics, 9 (2016), 1-20. Available at: http://scholarexchange.furman.edu/fuejum/vol9/iss1/1