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)?
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.