Consider the discrete-time coloring chain that chooses a uniform vertex and a uniform color from all q colors, accepting the proposed recoloring when it is proper and otherwise staying in place. On the four-vertex path, the worst-start total-variation mixing time at threshold 1/e equals 8 for q=17 and 9 for q=18. Thus increasing the number of colors can increase the mixing time even on a connected tree with no isolated vertices. We give an exact finite certificate, verified both by full-state integer enumeration and by a color-symmetry quotient. An elementary comparison with independent coordinate refresh also proves that the mixing time is 9 for every q>=512. The result concerns the literal proposal chain and a fixed finite graph, not an asymptotic slowdown on growing lattices. The previously reported edgeless example is credited. This AI-assisted, self-audited preprint is unrefereed and makes no absolute priority claim.
