  Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they
form a path (in this order) or a cycle of length three. An injective edge
coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that
if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq
c(e_3)$. The injective edge coloring number $\chi_{i}^{'}(G)$ is the minimum
number of colors permitted in such a coloring. In this paper, exact values of
$\chi_{i}^{'}(G)$ for several classes of graphs are obtained, upper and lower
bounds for $\chi_{i}^{'}(G)$ are introduced and it is proven that checking
whether $\chi_{i}^{'}(G)= k$ is NP-complete.
