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تسجيل مستخدم جديدWDM and Directed Star Arboricity
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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 $alpha$, $in(v,alpha)+out(v,alpha)leq n$ 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 $dst(D)leq 2Delta^-(D)+1$, and conjecture that if $Delta^-(D)geq 2$, then $dst(D)leq 2Delta^-(D)$. 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) tq D mbox{is $m$-labelled} et Delta^-(D)leq k}$. We show that if $mgeq n$, then $ds leftlceilfrac{m}{n}leftlceil frac{k}{n}rightrceil + frac{k}{n} rightrceilleq lambda_n(m,k) leqleftlceilfrac{m}{n}leftlceil frac{k}{n}rightrceil + frac{k}{n} rightrceil + C frac{m^2log k}{n}$ for some constant $C$. We conjecture that the lower bound should be the right value of $lambda_n(m,k)$.
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