# WDM and Directed Star Arboricity

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : 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 $\lambda$, $in(v,\lambda)+out(v,\lambda)\leq n$ 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 $v$ of label $l$ 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$.}$
Keywords :
Type de document :
Rapport
[Research Report] RR-6179, INRIA. 2007, pp.20
Domaine :

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https://hal.inria.fr/inria-00132396
Contributeur : Rapport de Recherche Inria <>
Soumis le : jeudi 3 mai 2007 - 10:36:41
Dernière modification le : jeudi 24 mai 2018 - 15:59:22
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 16:02:32

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• HAL Id : inria-00132396, version 3
• ARXIV : 0705.0315

### Citation

Omid Amini, Frédéric Havet, Florian Huc, Stéphan Thomassé. WDM and Directed Star Arboricity. [Research Report] RR-6179, INRIA. 2007, pp.20. 〈inria-00132396v3〉

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