Minimum number of moves in this Graph Coloring "Game"
The value of $p(G,v)$ is at most $r$ if you can label the vertices of $G$ with colors from $[v]^r$ in such a way that the colors assigned to every pair of adjacent vertices differ on at least one coordinate. Equivalently, $p(G,v) \leq r$ if you can label the vertices of $G$ with colors from $[v]^r$ in such a way that every pair of adjacent vertices gets a different color. Stated differently, $p(G,v) \leq r$ iff $\chi(G) \leq v^r$, where $\chi(G)$ is the chromatic number of $G$. This implies the formula
$$ p(G,v) = \lceil \log_v \chi(G) \rceil. $$
A similar relation should hold for the case of hypergraphs if you use the correct definition of chromatic number.