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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:

  1. 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.

  2. 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.

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It might be less confusing if you just claimed/asked if $\lfloor \frac{\Delta+3}2\rfloor$ colors suffice. However it is not correct. The $10$ edges of the complete graph $K_5$ can not be $3$ colored without creating a monochromatic $3$ edge path (which might be closed, i.e. a triangle.)

Here is a sketch which seems a little long but does the job:

  1. Note that $K_4$ can , as claimed, be colored with $3$ colors. However $2$ colors are not enough.

  2. In a possible $3$ edge coloring of $K_5$ observe that no vertex can touch $3$ (or $4$) edges of the same color because then the edges of the remaining vertices are restricted to the other two colors, and that contradicts 1.

  3. There are at least $4$ edges with the same color, say $a.$ Suppose $12,13$ are colored $a$. Then neither of $14,15$ is. Also none of $23,24,25,34,35$ can have color $a.$ But that leaves only $45$ which (so far) could have color $a$ and that is only three edges, not four.

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