  Motivated by investigations of rainbow matchings in edge colored graphs, we
introduce the notion of color-line graphs that generalizes the classical
concept of line graphs in a natural way. Let $H$ be a (properly) edge-colored
graph. The (proper) color-line graph $C\!L(H)$ of $H$ has edges of $H$ as
vertices, and two edges of $H$ are adjacent in $C\!L(H)$ if they are incident
in $H$ or have the same color. We give Krausz-type characterizations for
(proper) color-line graphs, and point out that, for any fixed $k\ge 2$,
recognizing if a graph is the color-line graph of some graph $H$ in which the
edges are colored with at most $k$ colors is NP-complete. In contrast, we show
that, for any fixed $k$, recognizing color-line graphs of properly edge colored
graphs $H$ with at most $k$ colors is polynomially. Moreover, we give a good
characterization for proper $2$-color line graphs that yields a linear time
recognition algorithm in this case.
