Analysis of a heuristic for acyclic edge colouring

An acyclic edge colouring of a graph is a proper edge colouring in which the union of any two colour classes does not contain a cycle, that is, forms a forest. It is known that there exists such a colouring using at most 16 Delta(G) colours where A(G) denotes the maximum degree of a graph G. However, no non-trivial constructive bound (which works for all graphs) is known except for the straightforward distance 2 colouring which requires Delta(2) colours. We analyse a simple O(mn Delta(2) (log Delta)(2)) time greedy heuristic and show that it uses at most 5 Delta(log z Delta + 2) colours on any graph. (c) 2006 Elsevier B.V. All rights reserved.