WDM and Directed Star Arboricity

A digraph is m-labelled if every arcs is labelled by an integer in {1, . . . , m}. Motivated by wavelength assignment for multicasts in optical star networks, we study n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour in λ, in(v, λ) + out(v, λ) ≤ n with in(v, λ) the number of arcs coloured λ entering v and out(v, λ) the number of labels l such that there exists an arc leaving v coloured λ. One likes to find the minimum number of colours λn(D) such that an m-labbelled digraph D has an n-fiber colouring. In the particular case, when D is 1-labelled then λn(D) is the directed star arboricty of D, denoted dst(D). We first show that dst(D) ≤ 2∆(D) + 1 and conjecture that if ∆(D) ≥ 2 then dst(D) ≤ 2∆(D). We also prove that if D is subcubic then dst(D) ≤ 3 and that if ∆(D), ∆(D) ≤ 2 then dst(D) ≤ 4. Finally, we study λn(m,k) = max{λn(D) | D is m-labelled and ∆ (D) ≤ k}. We show that if m ≥ n then ‰