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In this paper we study acyclic colouring in the random subgraph $\mathit{G}$of the complete graph $\mathit{K}_n$ on $\mathit{n}$ vertices where each edgeis present with probability $\mathit{p}$; independent of the other edges. Weshow that the acyclic chromatic number exhibits a phase transition fromsublinear to linear growth as the edge probability increases, even in thesparse regime and obtain estimates for the critical exponent. Next, weintroduce a relaxation by allowing for a small fraction of "bad" cycles toviolate the acyclic colouring condition and show that the critical exponent inthis case is in fact zero, no matter how small the fraction.

Critical Exponent for the Acyclic Chromatic Number of Random Graphs