Critical Exponent for the Acyclic Chromatic Number of Random Graphs

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 edge is present with probability $\mathit{p}$; independent of the other edges. We show that the acyclic chromatic number exhibits a phase transition from sublinear to linear growth as the edge probability increases, even in the sparse regime and obtain estimates for the critical exponent. Next, we introduce a relaxation by allowing for a small fraction of "bad" cycles to violate the acyclic colouring condition and show that the critical exponent in this case is in fact zero, no matter how small the fraction.