Backbone colorings along perfect matchings
Hajo J. Broersma, Jun Fujisawa, Kiyoshi Yoshimoto · University of Twente Research Information · 2003
Given a graph $G=(V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a backbone coloring for $G$ and $H$ is a proper vertex coloring $V\\rightarrow \\{1,2,\\ldots\\}$ of $G$ in which the colors assigned to adjacent vertices in $H$ differ by at least two. In a recent paper, backbone colorings were introduced and studied in cases were the backbone is either a spanning tree or a spanning path. Here we study the case where the backbone is a perfect matching. We show that for perfect matching backbones of $G$ the number of colors needed for a backbone coloring of $G$ can roughly differ by a multiplicative factor of at most $\\frac{4}{3}$ from the chromatic number $\\chi(G)$. We show that the computational complexity of the problem ``Given a graph $G$ with a perfect matching $M$, and an integer $\\ell$, is there a backbone coloring for $G$ and $M$ with at most $\\ell$ colors?'' jumps from polynomial to NP-complete between $\\ell=3$ and $\\ell=4$. Finally, we consider the case where $G$ is a planar graph.