,Let $G(V,E)$ be a weighted complete graph.

Let further $\min_k(v_i)$ denote, depending on whether the context is arithmetic or set theoretic, either the set of the $k$ smallest edges adjacent to $v_i$ or their weight-sum; with an overbar it denotes *half* the arithmetic mean, i.e. $\overline{\min_k(v)} := \frac{\min_k(v)}{2k}$

Question:Defining the following two steps

- for every vertex calculate $\pi(v) = -\overline{\min_k(v)}$ as its potential
- for every edge $e_{uv}$ set $w_{uv}=w_{uv}+\pi(u)+\pi(v)\ $ where $w_{uv}$ denotes the corresponding edge-weight
as an iteration, what will happen to the set $\cup_{v\in V}\min_k(v)$ of edges, if the iteration is carried out indefinitely

- will it converge unconditionally?
- if yes, will the graph induced by the limit set of edges in $\cup_{v\in V}\min_k(v)$ be $k$-regular, provided a k-regular spanner exists in $G$?