K-Equitable labeling

[math]\displaystyle{ k }[/math]-Equitable labeling --- [math]\displaystyle{ k }[/math]-справедливая разметка.

Let [math]\displaystyle{ G = (V,E) }[/math] be a graph. A labeling of [math]\displaystyle{ G }[/math] is a function [math]\displaystyle{ f: \; V \rightarrow \{0,1, \ldots, n\} }[/math] such that for each edge [math]\displaystyle{ e = (u,v) \in E }[/math], [math]\displaystyle{ f(e) = |f(u) - f(v)| }[/math]. Such a labeling is said to be [math]\displaystyle{ k }[/math]-equitable if it is a labeling of the vertices with the numbers [math]\displaystyle{ 0 }[/math] through [math]\displaystyle{ k - 1 }[/math] such that, if [math]\displaystyle{ v^{i} }[/math] is the number of vertices labeled [math]\displaystyle{ i }[/math], and [math]\displaystyle{ e^{i} }[/math] is the number of edges labeled [math]\displaystyle{ i }[/math], then [math]\displaystyle{ |v^{i} - v^{j}| \leq 1 }[/math] and [math]\displaystyle{ |e^{i} - e^{j}| \leq 1 }[/math] for all [math]\displaystyle{ i, j }[/math]. A graph is said to be [math]\displaystyle{ k }[/math]-equitable if it has [math]\displaystyle{ k }[/math]-equitable labeling.