WDM与定向星荫度
WDM and Directed Star Arboricity
摘要：
一个有向图是$m$ -标号的，如果在$\{1, \dots,m\}$中每个弧都用一个整数标号.受光网络中多播波长分配的启发，我们引入并研究了标记有向图的$n$ -纤维染色。这些是$D$的弧的着色，使得在每个顶点$v$处，并且对于每个颜色$α$，$in(v,α)+out(v,α)\leq n$具有$in(v,α)$，弧的数目着色为进入$v$的$α$，并且$out(v,α)$具有标记$l$的数目，使得存在至少一个标记$l$的弧离开$v$并且用$α$着色。问题是要找到最少的颜色数$λ_n(D)$，使得$m$标记的有向图$D$具有$n$纤维染色。在$D$被$1$标记的特定情况下，$λ_1(D)$被称为$D$的有向星荫度，并由$dst(D)$表示。我们首先证明了$dst(D)\leq 2Δ^-(D)+1$，并猜想，如果$Δ^-(D)\geq 2$，那么$dst(D)\leq 2Δ^-(D)$。我们还证明了，对于一个次立方有向图$D$，那么$dst(D)\leq 3$，并且，如果$Δ^+(D), Δ^-(D)\leq 2$，那么$dst(D)\leq 4$。最后，我们研究了$λ_n(m,k)=\max\{λ_n(D) \tq D \mbox{is $ m $-labelled} \et Δ^-(D)\leq k\}$。我们表明，如果$m\geq n$，那么$\ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq λ_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n}$为一些常数$C$。我们推测，下界应该是$λ_n(m,k)$的正确值。
英文摘要：
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 $α$, $in(v,α)+out(v,α)\leq n$ with $in(v,α)$ the number of arcs coloured $α$ entering $v$ and $out(v,α)$ the number of labels $l$ such that there is at least one arc of label $l$ leaving $v$ and coloured with $α$. The problem is to find the minimum number of colours $λ_n(D)$ such that the $m$-labelled digraph $D$ has an $n$-fibre colouring. In the particular case when $D$ is $1$-labelled, $λ_1(D)$ is called the directed star arboricity of $D$, and is denoted by $dst(D)$. We first show that $dst(D)\leq 2Δ^-(D)+1$, and conjecture that if $Δ^-(D)\geq 2$, then $dst(D)\leq 2Δ^-(D)$. We also prove that for a subcubic digraph $D$, then $dst(D)\leq 3$, and that if $Δ^+(D), Δ^-(D)\leq 2$, then $dst(D)\leq 4$. Finally, we study $λ_n(m,k)=\max\{λ_n(D) \tq D \mbox{is $m$-labelled} \et Δ^-(D)\leq k\}$. We show that if $m\geq n$, then $\ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq λ_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n}$ for some constant $C$. We conjecture that the lower bound should be the right value of $λ_n(m,k)$.
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