An edge colouring of a graph G is called acyclic if it is proper and every cycle contains at least three colours. We show that for every ε > 0, there exists a g = g(ε) such that if G has girth at least g then G admits an acyclic edge colouring with at most (1 + ε)∆ colours.

Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to an acyclic hamiltonian digraph. Our results include: ...

We give upper bounds for the generalised acyclic chromatic number and generalised acyclic edge chromatic number of graphs with maximum degree d, as a function of d. We also produce examples of graphs where these bounds are of the correct order.

We present some classes of graphs which satisfy the acyclic edge colouring conjecture which states that any graph can be acyclically edge coloured with at most ∆ + 2 colours.