%0 Journal Article
%T $P_3$-Rainbow Edge Colouring of Digraphs
%J Journal of Algorithms and Computation
%I University of Tehran
%Z 2476-2776
%D 2022
%\ 01/16/2022
%V
%N
%P 1-11
%! $P_3$-Rainbow Edge Colouring of Digraphs
%K planar digraphs
%K rainbow coloring
%K transitive digraph
%K dichromatic index
%R 10.22059/jac.2022.85517
%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_85517_65d76780f65db1c817b88f0d5705fed7.pdf