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WDM and directed star arboricity. (English) Zbl 1216.05044
Summary: A digraph is $$m$$-labelled if every arc is labelled by an integer in $$\{1,\dots, m\}$$. Motivated by wavelength assignment for multicasts in optical networks, we introduce and study $$n$$-fibre colourings of labelled digraphs. These are colourings of the arcs of $$D$$ such that at each vertex $$v$$, and for each colour $$\alpha$$, in$$(v, \alpha )$$ $$+$$ out$$(v, \alpha ) \leqslant n$$ with in$$(v, \alpha )$$ the number of arcs coloured $$\alpha$$ entering $$v$$ and out$$(v, \alpha )$$ the number of labels $$l$$ such that there is at least one arc of label $$l$$ leaving $$v$$ and coloured with $$\alpha$$.
The problem is to find the minimum number of colours $$\lambda _{n}(D)$$ such that the $$m$$-labelled digraph $$D$$ has an $$n$$-fibre colouring. In the particular case when $$D$$ is 1-labelled, $$\lambda _{1}(D)$$ is called the directed star arboricity of $$D$$, and is denoted by dst$$(D)$$. We first show that dst$$(D) \leqslant 2\Delta ^{ - }(D)+1$$, and conjecture that if $$\Delta ^{ - }(D) \geqslant 2$$, then dst$$(D) \leqslant 2\Delta ^{ - }(D)$$. We also prove that for a subcubic digraph $$D$$, then dst$$(D) \leq 3$$, and that if $$\Delta ^{+}(D), \Delta ^{ - }(D) \leq 2$$, then dst$$(D) \leq 4$$.
Finally, we study $$\lambda _{n}(m, k) = \max\{\lambda _{n}(D) | D$$ is $$m$$-labelled and $$\Delta ^{ - }(D) \leq k\}$$. We show that if $$m \geqslant n$$, then $$\lceil \frac m n \lceil \frac k n \rceil +\frac k n \rceil \leqslant \lambda _n (m,k)\leqslant \lceil \frac m n \lceil \frac k n \rceil +\frac k n \rceil + C\frac{m^2 \log k}{n}$$ for some constant $$C$$. We conjecture that the lower bound should be the correct value of $$\lambda _n (m,k)$$.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
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