WDM and Directed Star Arboricity

Description

A digraph is {\it $m$-labelled} if every arcs is labelled by an integer in $\{1, \dots ,m\}$. Motivated by wavelength assignment for multicasts in optical star networks, we study {\it $n$-fiber colourings} of labelled digraph which are colourings of the arcs of $D$ such that at each vertex $v$, for each colour in $\lambda$, $in(v,\lambda)+out(v,\lambda)\leq n$ with $in(v,\lambda)$ the number of arcs coloured $\lambda$ entering $v$ and $out(v,\lambda)$ the number of labels $l$ such that there exists an arc leaving $v$ coloured $\lambda$. One likes to find the minimum number of colours $\lambda_n(D)$ such that an $m$-labbelled digraph $D$ has an $n$-fiber colouring. In the particular case, when $D$ is $1$-labelled then $\lambda_n(D)$ is the {\it directed star arboricty} of $D$, denoted $dst(D)$. We first show that $dst(D)\leq 2\Delta^-(D)+1$ and conjecture that if $\Delta^-(D)\geq 2$ then $dst(D)\leq 2\Delta^-(D)$. We also prove that if $D$ is subcubic 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) \tq D \mbox{ is $m$-labelled} \et \Delta^-(D)\leq k\}$. We show that if $m\geq n$ then $\ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq \lambda_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n} \mbox {\ \ \ \ \ for some constant $C$.}$

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