Oscillation of a graph

Oscillation of a graph --- осцилляция графа.

An edge-ordering of the finite simple graph [math]\displaystyle{ G = (V,E) }[/math] is 1-1 function [math]\displaystyle{ f }[/math] from [math]\displaystyle{ E }[/math] to the set of positive integers. The set of all edge-orderings of [math]\displaystyle{ G }[/math] is denoted by [math]\displaystyle{ {\mathcal F} }[/math]. For [math]\displaystyle{ f \in {\mathcal F} }[/math], a path with the edge sequence [math]\displaystyle{ e_{1}, e_{2}, \ldots, e_{t} }[/math] is called an [math]\displaystyle{ f-zpath }[/math] if for each [math]\displaystyle{ i = 1, \ldots, t-2 }[/math]

[math]\displaystyle{ f(e_{i}) - f(e_{i+1}) \gt 0\mbox{ if and only if } f(e_{i+1}) - F(e_{i+2}) \gt 0. }[/math]

The vibration [math]\displaystyle{ k(f) }[/math] of [math]\displaystyle{ f }[/math] is the maximum length of an [math]\displaystyle{ f-z path }[/math] and the oscillation [math]\displaystyle{ \eta(G) }[/math] of [math]\displaystyle{ G }[/math] is defined by

[math]\displaystyle{ \eta(G) = \min_{f \in} k(f). }[/math]

Observe that [math]\displaystyle{ \eta(G) }[/math] is the greatest integer [math]\displaystyle{ t }[/math] such that [math]\displaystyle{ G }[/math] has [math]\displaystyle{ f-zpath }[/math] for each [math]\displaystyle{ f \in {\mathcal F} }[/math].