Acyclic colouring of graphs

A vertex colouring of a graph G is called acyclic if no two adjacent vertices have the same colour and there is no two-coloured cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colours in an acyclic colouring of G. We show that if G has maximum degree d then A(G) = O(d 4 3 ) as d → ∞. This settles a problem of Erdős who conjectured, in 1976, that A(G) = o(d2) as d →∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d 4 3 /(log d) 1 3 ); and that the edges of any graph with maximum degree d can be coloured by O(d) colours so that no two adjacent edges have the same colour and there is no two-coloured cycle. All the proofs rely heavily on probabilistic arguments.