Colouring Edges with many Colours in Cycles
Abstract
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 Arb_p(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, Arb_p(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 the multigraphs having a shallow subdivision as a subgraph of G.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2011
- DOI:
- 10.48550/arXiv.1108.1616
- arXiv:
- arXiv:1108.1616
- Bibcode:
- 2011arXiv1108.1616N
- Keywords:
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- Mathematics - Combinatorics