  We study a family of closely-related distributed graph problems, which we
call degree splitting, where roughly speaking the objective is to partition (or
orient) the edges such that each node's degree is split almost uniformly. Our
findings lead to answers for a number of problems, a sampling of which
includes:
  -- We present a $poly(\log n)$ round deterministic algorithm for
$(2\Delta-1)\cdot (1+o(1))$-edge-coloring, where $\Delta$ denotes the maximum
degree. Modulo the $1+o(1)$ factor, this settles one of the long-standing open
problems of the area from the 1990's (see e.g. Panconesi and Srinivasan
[PODC'92]). Indeed, a weaker requirement of $(2\Delta-1)\cdot poly(\log
\Delta)$-edge-coloring in $poly(\log n)$ rounds was asked for in the 4th open
question in the Distributed Graph Coloring book by Barenboim and Elkin.
  -- We show that sinkless orientation---i.e., orienting edges such that each
node has at least one outgoing edge---on $\Delta$-regular graphs can be solved
in $O(\log_{\Delta} \log n)$ rounds randomized and in $O(\log_{\Delta} n)$
rounds deterministically. These prove the corresponding lower bounds by Brandt
et al. [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight.
Moreover, these show that sinkless orientation exhibits an exponential
separation between its randomized and deterministic complexities, akin to the
results of Chang et al. for $\Delta$-coloring $\Delta$-regular trees.
  -- We present a randomized $O(\log^4 n)$ round algorithm for orienting
$a$-arboricity graphs with maximum out-degree $a(1+\epsilon)$. This can be also
turned into a decomposition into $a (1+\epsilon)$ forests when $a=\Omega(\log
n)$ and into $a (1+\epsilon)$ pseduo-forests when $a=o(\log n)$. Obtaining an
efficient distributed decomposition into less than $2a$ forests was stated as
the 10th open problem in the book by Barenboim and Elkin.
