### 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.

Original language | English |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Publication status | Published - 2003 |

### Publication series

Name | Memorandum |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1706 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-05C85
- MSC-05C15
- IR-65891
- EWI-3526
- MSC-05C17

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## Cite this

Broersma, H. J., Fujisawa, J., & Yoshimoto, K. (2003).

*Backbone colorings along perfect matchings*. (Memorandum; No. 1706). Enschede: University of Twente, Department of Applied Mathematics.