Color-line and Proper Color-line Graphs

TL;DR

This paper introduces color-line graphs, generalizing line graphs with edge coloring, provides characterizations, and analyzes the computational complexity of recognizing such graphs under various coloring constraints.

Contribution

It defines color-line graphs, offers Krausz-type characterizations, and establishes complexity results for recognizing these graphs with fixed color bounds.

Findings

Recognition of color-line graphs with at most k colors is NP-complete for general graphs.

Recognition of properly edge-colored color-line graphs with at most k colors is polynomial.

Linear time recognition algorithm for proper 2-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 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.