Magic labeling

Magic labeling --- магическая разметка.

Magic labeling is one-to-one map onto the appropriate set of consecutive integers starting from 1, satisfying some kind of constant-sum pro\-perty. A vertex-magic labeling is one in which the sum of all labels associated with a vertex is a constant independent of the choice of the vertex. Edge-magic labelings are defined similarly. Vertex-magic total labeling is a one-to-one mapping

[math]\displaystyle{ \lambda: \; E \cup V \rightarrow \{1, 2, \ldots, |V|+|E|\} }[/math]

with the property that there is a constant [math]\displaystyle{ k }[/math] such that at any vertex [math]\displaystyle{ x }[/math]

[math]\displaystyle{ \lambda(x) + \sum\lambda(xy) = k }[/math]

where the sum is over all vertices [math]\displaystyle{ y }[/math] adjacent to [math]\displaystyle{ x }[/math]. For any labeling we call the sum of the appropriate labels at a vertex the weight of the vertex, denoted [math]\displaystyle{ wt(x) }[/math]; so for vertex-magic total labelings we require that the weight of all vertices be the same, namely [math]\displaystyle{ k }[/math] and this number is called the magic constant for the labeling.