@book{76ad6a22f28d45379f60dba3ae8ce037,

title = "Backbone colorings along perfect matchings",

abstract = "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.",

keywords = "MSC-05C85, MSC-05C15, IR-65891, EWI-3526, MSC-05C17",

author = "Broersma, {Haitze J.} and J. Fujisawa and K. Yoshimoto",

year = "2003",

language = "English",

series = "Memorandum",

publisher = "University of Twente, Department of Applied Mathematics",

number = "1706",

}