This question is related to my previous post: New edge coloring problem in graph theory.

*Added:* 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 $s$.

Obviously $s\leq \Delta$. Is the following clime true? Could someone provide a counterexample or a sketch of proof for this clime?

Clime (Edited after Chris Godsil Answer):Edges of every simple graph can be colored due to above coloring with at most $s+1$ color. where $s=\lfloor\frac{\Delta+1}{2}\rfloor$+1.