Menu
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 \( \{1,2,\ldots,|V| + |E|\} \) 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 \( \{1, 2,\ldots,|E|\} \) 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.