Colouring edges with many colours in cycles

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular case where G has girth at least p + 1, Arbp(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring be proper, i.e., adjacent edges receive distinct colours, then the minimum number of colours needed is the generalized p-acyclic edge chromatic number of G. In this paper, we relate the generalized p-acyclic edge chromatic numbers and the generalized p-arboricities of a graph G to the density of ∗Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI) Charles University Malostranské nám.25, 11800 Praha 1, Czech Republic, email:[email protected] Supported by grant 1M0405 of the Czech Ministry of Education †Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557) 190-198 avenue de France, 75013 Paris, France, email:[email protected] Partially supported by the Academia Sinica ‡Department of Mathematics Zhejiang Normal University, China email:[email protected]