WDM and Directed Star Arboricity

Abstract

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 $λ_n(m,k)$.