[hansvm]

Suppose the graph admits only 4+ colorings, but when attempting a 3-coloring it's possible for only one edge to be misaligned. Then (A) you need O(n_edges) calls to the oracle to gain any confidence about the 3-colorability of a 3-colorable graph (else you might be easily duped by the one misaligned edge), and (B) in so doing, you learn almost all of the structure of the graph (since you have way more random calls than there are edges).

Restating, not only is the ZK algorithm slow, but by the time you have confidence in the ZK proof you also have additional knowledge about the structure whose properties you're proving.

[ColinWright]

In the version I'm discussing, both parties already know the graph, so your point (B) seems irrelevant. The interrogator needs to know the graph in order to specify an edge (and to know that they've done so), so I'm not sure of the precise version you are thinking of.

Put point (A) is relevant ... certainly each call only provides a small amount of additional confidence, so a lot of calls might be required. Even so, the system seems sound to me, and I'd appreciate any details of ways in which it is not.

See also my comment here: https://news.ycombinator.com/item?id=46121137

That gives more detail of the setup, and how it can be implemented digitally.