Let $G$ be a simple graph. Consider the following edge coloring:

We are allowed to use repetitive colors on some edges incident to a vertex such that the result does not contain a sequence of length $3$ of one color.

The maximum different colors used for coloring the edges incident to a vertex is $s< \Delta(G)$.

Question 1:Is Question 2 a known graph theory problem?

Question 2:what is the smallest number of colors needed to color the edges of $G$ according to (1) and (2)?

Update.

**Remark:** Note that if $s=\Delta$ and without repetitive color then the problem reduces to usual edge coloring. this is useful for understanding this problem.