%0 Journal Article %T Rainbow Edge Colouring of Digraphs %J Journal of Algorithms and Computation %I University of Tehran %Z 2476-2776 %A Hasheminezhad, Mahdieh %D 2021 %\ 12/01/2021 %V 53 %N 2 %P 165-172 %! Rainbow Edge Colouring of Digraphs %K vertex equitable labeling %K vertex rainbow coloring %K planar digraphs %K template-driven rainbow coloring %K transitive digraph %K dichromatic index %R 10.22059/jac.2021.85350 %X An edge  coloring of a digraph  $D$ is called a $P_3$-rainbow edge coloring if  the edges of any directed path of $D$ with length 2 are colored with different colors. It is proved that  for a $P_3$-rainbow edge coloring of  a digraph $D$, at least $\left\lceil{log_2{\chi(D)}} \right\rceil$ colors are necessary and $ 2\left\lceil{log_2{\chi(D)}}\right\rceil\}$  colors are enough. One can determine in linear time if  a digraph has a  $P_3$-rainbow edge coloring with 1 or 2 colors. In this paper, it is proved that  determining   that a digraph has a  $P_3$-rainbow edge coloring  with 3 colors is an NP-complete problem even for planar digraphs. Moreover, it is shown that  $\left\lceil{log_2{\chi(D)}}\right\rceil$ colors is necessary and sufficient for a $P_3$-rainbow edge coloringof a transitive orientation digraph $D$. %U https://jac.ut.ac.ir/article_85350_6b9c10b49ca98e2833bb83a0b59ee563.pdf