On Completely k-Magic Regular Graphs

Let k be a positive integer. A graph G = (V (G), E(G)) is said to be k-magic if there is a function (or edge labeling) ` : E(G) → Zk \ {0}, where Z1 = Z, such that the induced function (or vertex labeling) ` + : V (G) → Zk, defined by ` +(v) = P uv∈E(G) `(uv), is a constant map, where the sum is taken in Zk. We say that G is c-sum k-magic if ` +(v) = c for all v ∈ V (G). The set of all c ∈ Zk such that G is c-sum k-magic is called the sum spectrum of G with respect to k. In the case when the sum spectrum of G is Zk, we say that G is completely k-magic. In this paper, we determine all completely 1-magic regular graphs. After observing that any 2-magic graph is not completely 2-magic, we show that some regular graphs are completely k-magic for k ≥ 3, and determine the sum spectra of some regular graphs that are not completely k-magic

Characterization of completely k-magic regular graphs

Let k ∈ N and c ∈ Zk. A graph G is said to be c-sum k-magic if there is a labeling ` : E(G) → Zk \ {0} such that P u∈N(v) `(uv) ≡ c (mod k) for every vertex v of G, where N(v) is the neighborhood of v in G. We say that G is completely k-magic whenever it is c-sum k-magic for every c ∈ Zk. In this paper, we characterize all completely k-magic regular graphs

Vertex Magic Group Edge Labelings

A project submitted to the faculty of the graduate school of the University of Minnesota in partial fulfillment of the requirements for the degree of Master of Science. May 2017. Major: Mathematics and Statistics. Advisor: Dalibor Froncek. 1 computer file (PDF); vi, 46 pages, appendix A, Ill. (some col.)A vertex-magic group edge labeling of a graph G(V;E) with |E| = n is an injection from E to an abelian group ᴦ of order n such that the sum of labels of all incident edges of every vertex x ϵ V is equal to the same element µ ϵ ᴦ. We completely characterize all Cartesian products Cn□Cm that admit a vertex-magic group edge labeling by Z2nm, as well as provide labelings by a few other finite abelian groups