  The local chromatic number is a coloring parameter defined as the minimum
number of colors that should appear in the most colorful closed neighborhood of
a vertex under any proper coloring of the graph. Its directed version is the
same when we consider only outneighborhoods in a directed graph. For digraphs
with all arcs being present in both directions the two values are obviously
equal. Here we consider oriented graphs. We show the existence of a graph where
the directed local chromatic number of all oriented versions of the graph is
strictly less than the local chromatic number of the underlying undirected
graph. We show that for fractional versions the analogous problem has a
different answer: there always exists an orientation for which the directed and
undirected values coincide. We also determine the supremum of the possible
ratios of these fractional parameters, which turns out to be e, the basis of
the natural logarithm.
