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Register a new userWDM and Directed Star Arboricity
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Added by
Omid Amini
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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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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We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the following three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Watanabe conjectured that every bipartite planar graph has an induced forest on at least five-eighths of its vertices, and in 2010, Kowalik, Luv{z}ar, and v{S}krekovski conjectured that every planar graph of girth at least five has an induced forest on at least seven-tenths of its vertices. We make progress toward the fractional generalization of the latter of these, by proving that every planar graph of girth at least five has fractional vertex-arboricity at most $2 - 1/324$.
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