TY - JOUR ID - 85517 TI - $P_3$-Rainbow Edge Colouring of Digraphs JO - Journal of Algorithms and Computation JA - JAC LA - en SN - 2476-2776 Y1 - 2022 PY - 2022 VL - IS - SP - 1 EP - 11 KW - planar digraphs KW - rainbow coloring KW - transitive digraph KW - dichromatic index DO - 10.22059/jac.2022.85517 N2 - 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 $D$.  UR - https://jac.ut.ac.ir/article_85517.html L1 - https://jac.ut.ac.ir/article_85517_65d76780f65db1c817b88f0d5705fed7.pdf ER -