Vertex-distinguishing proper arc colorings of digraphs

An arc coloring of a digraph DD is proper if (i) no two arcs with a common tail receive the same color and (ii) no two arcs with a common head receive the same color. Define the out-arc set and in-arc set   of a vertex vv of DD to be the set of arcs with tail vv and the set of arcs with head vv, respectively. A proper arc coloring of DD is vertex-distinguishing (resp. semi-vertex-distinguishing  ) if (i) no two vertices (resp. no three vertices) have the same color set for their in-going arcs and (ii) no two vertices (resp. no three vertices) have the same color set for their out-going arcs. And a proper arc coloring of DD is equitable if the numbers of arcs colored by any two colors differ by at most one.In this paper, (semi-)vertex-distinguishing proper arc colorings of digraphs are introduced. Denote by χvd′(D) (resp. χsvd′(D)) the minimum number of colors required for a vertex-distinguishing (resp. semi-vertex-distinguishing) proper arc coloring of DD. We give upper bounds for χvd′(D) and χsvd′(D) respectively. In particular, the value of χvd′(D) is obtained for some regular digraphs DD. Moreover, we show that the values of χvd′(D) and χsvd′(D) will not be changed if the coloring is, in addition, required to be equitable.