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}