In zero knowledge, in the context of graph 3-coloring, I do not understand how the prover is actually showing that he solved the problem.
both know the graph (the prover and the verifier)
the prover solves a 3-coloring of the graph
*the prover randomly permutes the colors
*the verifier asks for the coloring of an edge
*if the colors are different the process continues, and the verifier is more convinced that the prover actually solved the 3-coloring, if not, the verifier knows that the prover did not solve the 3-coloring
My question is: why couldn't a false prover (someone that does not know how to solve for the coloring) just always provide 2 different random colors?