Neighbor-Distinguishing Arc Colorings of Several Digraphs

If there is a choice function \(f{:}A(D) \to \left[ k \right]\) such that any arc with a common starting vertex and any arc with a common ending vertex do not have the same color in digraph \(D\) , then \(f\) is a proper arc coloring. Let \(S^{ - } (u)\) and \(S^{ + } (u)\) denote by the color sets of the incoming arcs and the outgoing arcs of \(u\) separately. If there is a proper arc coloring \(f\) such that the color sets of the outgoing arcs of \(u\) are not as same as the color sets of the incoming arcs of \(v\) for every arc \(\overrightarrow {uv} \in A(D)\) , then \(f\) is neighbour-distinguishing. The neighbor-distinguishing arc coloring \(f\) of \(D\) requires the minimum arc chromatic number to be denoted as \(\chi^{\prime}_{ + ,- } (D)\) . In this article, we prove the upper bound of a general digraph and the exact results of neighbor-distinguishing arc chromatic number of several special digraph.