Sure, it's allowed, and makes sense in this particular example. But someone reading this might think that the scheme depends on using a random coloring in the s

Sure, it's allowed, and makes sense in this particular example. But someone reading this might think that the scheme depends on using a random coloring in the second round, which is precisely not the case. In ZKP, usually the claim is "I have constructed this 3-colorable graph, and I wish to prove to you that it is 3-colorable without giving you any info about the coloring", and I only know the one colorability that I constructed the graph with (so the narrative given about Google calculating a coloring for you isn't the best analogy to begin with). Or, "Here's a number that's a product of two big primes, a fact which I will prove to you without giving you a hint on what the primes are," in which there is obviously only one solution.