Vertex Magic Total Labeling of Products of Regular VMT Graphs and Regular Supermagic Graphs

Abstract

A vertex-magic total labeling of a graph $G(V,E)$ is defined as a one-to-one mapping from $V\cup E$ to the set of integers $\{1,2,\dots ,|V|+|E|\}$ with the property that the sum of the label of a vertex and the labels of all edges incident to this vertex is the same constant for all vertices of the graph. A supermagic labeling of a graph $G(V,E)$ is defined as a one-to-one mapping from $E$ to the set of integers $\{1,2,\dots ,|E|\}$ with the property that the sum of the labels of all edges incident to a vertex is the same constant for all vertices of the graph.

In this paper, we present a technique for constructing vertex-magic total labelings of products of certain vertex-magic total $r$-regular graphs $G$ and certain $2_s$-regular supermagic graphs $H$. $H$ has to be decomposable into two $s$-regular factors and if $r$ is even, $|H|$ has to be odd.