Color-line and proper color-line graphs

Abstract

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 graphCL(H) of H has edges of H as vertices, and two edges of H are adjacent in CL(H) if they have an endvertex in common or have the same color.We give Krausz-type characterizations for (proper) color-line graphs, and show that, for any fixed k, recognizing color-line graphs of properly edge colored graphs H with at most k colors is polynomially solvable. Moreover, we give a good characterization for proper 2-color-line graphs that yields a linear time recognition algorithm in this case.In contrast, we 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.