Colouring Edges with many Colours in Cycles

Jaroslav Nesetril (1) , Patrice Ossona de Mendez (2) , Xuding Zhu (3)

1 KAM - Department of Applied Mathematics (KAM)
2 CAMS - Centre d'Analyse et de Mathématique sociales
3 Department of Mathematics [Hangzhou]

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.

Domaines: Combinatoire [math.CO]

Jaroslav Nesetril, Patrice Ossona de Mendez, Xuding Zhu. Colouring Edges with many Colours in Cycles. Journal of Combinatorial Theory, Series B, 2014, 109, pp.102-119. ⟨10.1016/j.jctb.2014.06.002⟩. ⟨hal-00613907⟩