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Abstract:
We give a distributed randomized algorithm to edge color a
network. Given a graph $G$ with $n$ nodes and maximum degree
$\Delta$, the algorithm,
\begin{itemize}
\item For any fixed $\lambda >0$, colours $G$ with $(1+ \lambda)
\Delta$ colours in time $O(\log n)$.
\item For any fixed positive integer $s$, colours $G$ with
$\Delta + \frac {\Delta} {(\log \Delta)^s}=(1 + o(1)) \Delta $
colours in time $O (\log n + \log ^{2s} \Delta \log \log
\Delta $.
\end{itemize}
Both results hold with probability arbitrarily close to 1
as long as $\Delta (G) = \Omega (\log^{1+d}
n)$, for some $d>0$.\\
The algorithm is based on the R"odl Nibble, a probabilistic strategy
introduced by Vojtech R"odl. The analysis involves a certain
pseudo--random phenomenon involving sets at the
vertices
