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arXiv · 1511.01025

Color-line and Proper Color-line Graphs

Van Bang Le·Florian Pfender
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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.

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Van Bang Le, Florian Pfender. 2015-11-03. Color-line and Proper Color-line Graphs. https://arxiv.org/abs/1511.01025

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