Acyclic Edge Colouring of Partial 2-Trees

An acyclic edge colouring of a graph is a proper edge colouring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and it is denoted by a′(G). Here, we obtain tight estimates on a′(G) for nontrivial subclasses of the family of 2-degenerate graphs. Specifically, we obtain values of the acyclic chromatic index for the families of partial 2-trees and series-parallel graphs. Another family contained within the family of 2-degenerate graphs is the family of outerplanar graphs. The acyclic chromatic index for outerplanar graphs has already been tightly determined. It was conjectured by Alon, Sudakov and Zaks that a′(G) ≤ ∆+ 2, where ∆ = ∆(G) denotes the maximum degree of the graph. Here we verify the conjecture for the classes of graphs considered. We first prove that a′(G) ≤ ∆+ 1 for partial 2-trees. As a corollary, it follows that the same bound holds for the class of series-parallel graphs, as it is a subclass of the class of partial 2-trees. These are best possible upper bounds, as there are examples graphs in the corresponding families which require that many colours for an acyclic edge colouring. All these bounds are proved constructively, leading to effecient algorithms to produce colourings with the stated number of colours.