21787 articles – 15600 references  [version française]
 HAL: inria-00132396, version 2
 arXiv: 0705.0315
 Available versions v1 (2007-02-21) v2 (2007-05-02) v3 (2007-05-03)
 WDM and Directed Star Arboricity
 Omid Amini 1, Frederic Havet 1
 (2007)
 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$.}$
 1: MASCOTTE (INRIA Sophia Antipolis / Laboratoire I3S) INRIA – Université Nice Sophia Antipolis [UNS] – CNRS : UMR7271 2: Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM) CNRS : UMR5506 – Université Montpellier II - Sciences et techniques
 Research team (except Inria): INFO/ALGCO
 Domain : Computer Science/Networking and TelecommunicationMathematics/Combinatorics
 Keywords: WDM optical networks – multicasting – graph colouring – complexity
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 inria-00132396, version 2 http://hal.inria.fr/inria-00132396 oai:hal.inria.fr:inria-00132396 From: Omid Amini <> Submitted on: Wednesday, 2 May 2007 17:47:55 Updated on: Wednesday, 2 May 2007 17:55:09