Is There An Ideal Color Wheel?

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors.
One might wonder: is there a color wheel in which all six of the sections have this property, without all the sections being the same color? We show that the answer is no, not just for the 6-cycle but for any finite connected graph; indeed, for any finite, strongly connected, edge-weighted digraph. The result generalizes the ``harmonic lemma" for graphs, replacing the well-behaved averaging function by paint blending, about which almost nothing is assumed. Our proof makes use of a Markov chain stopping rule.