Is there a planar 4-regular graph that is 3-acyclic colourable?
I can prove that no 4-regular graphs are 3-acyclic colorable.

Consider a 4-regular graph with a 3-coloring. If we call the colors $a, b, c$, then one of the three subgraphs generated by restricting to either $a$ and $b$ colored vertices, $a$ and $c$ vcertices, or $b$ and $c$ vertices must have as many edges as vertices. But all graphs with as many edges as vertices contain a cycle, so the coloring is not acyclic.

Let the graph have $k$ vertices. The graph must have $2k$ edges, because it is 4-regular. Let the number of $a$ colored vertices be $k_a$, and similarly for $k_b$ and $k_c$. Call $e_{ab}$ the number of edges between $a$ and $b$ colored vertices, and the same for $e_{ac}$ and $e_{bc}$. If we assume that none of the restricted graphs have as many edges as vertices, we must have
$$
e_{ab} &lt; k_a + k_b \\
e_{ac} &lt; k_a + k_c \\
e_{bc} &lt; k_b + k_c \\
$$
Summing over all edges, we find that $e_{ab} + e_{ac} + e_{bc} &lt; 2(k_a + k_b + k_c)$, and so $e &lt; 2k$. But this contradicts the fact that 4-regular graphs have twice as many edges as vertices, mentioned above. Thus, the assumption is false, and some color-restricted subgraph has as many edges as vertices, and hence has a cycle.