Hardness Transitions and Uniqueness of Acyclic Colouring

For $k\in \mathbb{N}$, a $k$-acyclic colouring of a graph $G$ is a function $f\colon V(G)\to \{0,1,\dots,k-1\}$ such that (i)~$f(u)\neq f(v)$ for every edge $uv$ of $G$, and (ii)~there is no cycle in $G$ bicoloured by $f$. For $k\in \mathbb{N}$, the problem $k$-ACYCLIC COLOURABILITY takes a graph $G$ as input and asks whether $G$ admits a $k$-acyclic colouring. Ochem (EuroComb 2005) proved that 3-ACYCLIC COLOURABILITY is NP-complete for bipartite graphs of maximum degree~4. Mondal et al. (J. Discrete Algorithms, 2013) proved that 4-ACYCLIC COLOURABILITY is NP-complete for graphs of maximum degree five. We prove that for $k\geq 3$, $k$-ACYCLIC COLOURABILITY is NP-complete for bipartite graphs of maximum degree $k+1$, thereby generalising the NP-completeness result of Ochem, and adding bipartiteness to the NP-completeness result of Mondal et al. In contrast, $k$-ACYCLIC COLOURABILITY is polynomial-time solvable for graphs of maximum degree at most $0.38\, k^{\,3/4}$. Hence, for $k\geq 3$, the least integer $d$ such that $k$-ACYCLIC COLOURABILITY in graphs of maximum degree $d$ is NP-complete, denoted by $L_a^{(k)}$, satisfies $0.38\, k^{\,3/4}<L_a^{(k)}\leq k+1$. We prove that for $k\geq 4$, $k$-ACYCLIC COLOURABILITY in $d$-regular graphs is NP-complete if and only if $L_a^{(k)}\leq d\leq 2k-3$. We also show that it is coNP-hard to check whether an input graph $G$ admits a unique $k$-acyclic colouring up to colour swaps (resp. up to colour swaps and automorphisms).