Color-line and Proper Color-line Graphs

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 CL(H) of H has edges of H as vertices, and two edges of H are adjacent in CL(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 ≥ 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.