How to verify Hamilton-Path in log-space? Given an undirected graph $G$ and an undirected path $p$, Is it possible to verify $p$ is a Hamilton path in graph $G$ using logarithmic space? How is it possible to verify the path goes through all vertices without somehow save which one had been already checked?
You can check all vertices in order.

Suppose we are given vertices $v_1, v_2, \cdots, v_n$ and a path $p=(p_1,p_2, \cdots,p_m)$.Check if $m==n$. If not, reject. Otherwise, check if $v_1$ appears in $p$. If not, reject.  Otherwise, check if $v_2$ appears in $p$. If not reject. And so on. Finally check if $v_m$ appears in $p$. If not, reject. Otherwise, accept.