So I think I understand how zero knowledge protocol with 3-coloring is supposed to work. But in an attempt to increase soundness of the protocol, we allow the verifier V to challenge two edges per repetition instead of one. I'm told that this breaks the zero knowledge property, but I'm not exactly sure how to prove it.
How would a malicious verifier V that repeats this modified zero knowledge protocol a polynomial number of times be able to construct an honest P's entire witness? In other words, how could V learn to color the entire graph when it can open two edges at a time when one edge at a time doesn't work? And why does it have to br a malicious verifier that breaks the property and not an honest verifier?