This is a follow-up to [this question](https://mathoverflow.net/questions/285232/different-form-of-edge-coloring).


Let $G=(V,E)$ be a finite, simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. More formally, we want the restriction $c|\_{E(v)}$ to be non-constant, where $E(v) = \{e\in E: x\in e\}$. (Is there a name for this kind of edge-coloring?)


Is there some positive integer $n\_0\in\mathbb{N}$ such every graph $G=(V,E)$ with minimal degree $2$ has a weak edge coloring $c:E\to\{1,\ldots,n\_0\}$? If yes, what is the smallest such integer?



Upper bound for weak edge colorings?
According to the answers to the previous question, the only connected graphs which require more than two colors are odd cycles, which are trivially weak 3-colorable. Therefore, $n\_0 = 3$ always suffices.


