Document Type : Research Paper


Department of Computer Science, Combinatorial and Geometric Algorithms Lab,Yazd University, Yazd, Iran


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 coloring
of a transitive orientation digraph $D$.