Suppose I nondeterministically walk around in a graph with n vertices.
When looking for a Hamiltonian path, at some point I’ve walked n/2 vertices.
There are (n choose n/2) different combinations of vertices I could have walked (meaning the unordered set, not the ordered walk).
Each of those states must be distinct from one another.
If not, then, depending on the remaining n/2 vertices, I would decide the wrong answer.
Therefore, midway through the path, at n/2, I need (n choose n/2) different states. That is too big for logspace.
Therefore you cannot decide a Hamiltonian path by nondeterministically walking around.
Does this imply Hamiltonian path cannot be decided in nondeterministic logspace - at least by "walking around"?