Vertex-magic total labeling

Vertex-magic total labeling --- вершинно-магическая тотальная разметка.

A one-to-one map [math]\displaystyle{ \lambda }[/math] from [math]\displaystyle{ E \cup V }[/math] onto integers [math]\displaystyle{ \{1,2, \ldots, e+v\} }[/math] is a vertex-magic total labeling, if there is a constant [math]\displaystyle{ k }[/math] such that for every 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]. Let us call the sum of labels at the vertex [math]\displaystyle{ x }[/math] the weight of the vertex; we require [math]\displaystyle{ wt(x) = k }[/math] for all [math]\displaystyle{ x }[/math]. The constant [math]\displaystyle{ k }[/math] is called the magic constant for [math]\displaystyle{ \lambda }[/math]. The edge labels are all distinct.