6-coloring of a tree in a distributed manner
In the paper, the authors mention that after $\log^* n$ iterations, the number of bits used to encode any of the color labels is at most 3. That means that there are at most eight colors. If we have six or fewer colors, we're done, and otherwise we go through the loop once more.

Let's say that we do one last loop iteration. The root gets the color label 0 or 1. Every other node gets a label equal to the first bit position where that node's old label disagrees with its parent's label, plus one extra bit. Since on entry to this iteration all colors are three bits long in the worst case, there are three possible positions where the bits could disagree. Therefore, each new color is formed from a combination of one of three choices and a bit (one of two choices), so at exit of this loop iteration there are only six possible color labels, so we have a six-coloring. The total number of loop iterations is then $\log^* n + 1 = O(\log^* n)$

Hope this helps!