### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 143-162 |

Number of pages | 20 |

Journal | Discussiones mathematicae. Graph theory |

Volume | 29 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2009 |

### Keywords

- EWI-15830
- METIS-263962
- IR-67842

### Cite this

*Discussiones mathematicae. Graph theory*,

*29*(1), 143-162. https://doi.org/10.7151/dmgt.1437

}

*Discussiones mathematicae. Graph theory*, vol. 29, no. 1, pp. 143-162. https://doi.org/10.7151/dmgt.1437

**Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number.** / Broersma, Haitze J.; Marchal, Bert; Paulusma, Daniël; Salman, M.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

AU - Broersma, Haitze J.

AU - Marchal, Bert

AU - Paulusma, Daniël

AU - Salman, M.

PY - 2009/3

Y1 - 2009/3

N2 - We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph $G = (V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\gamma$-backbone coloring for $G$ and $H$ is a proper vertex coloring $V\to \{1,2,\ldots\}$ of $G$ in which the colors assigned to adjacent vertices in $H$ differ by at least $\gamma$. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number $\ell$ of colors, for which such colorings $V\to \{1,2,\ldots, \ell\}$ exist, in the worst case is a factor times the chromatic number (for path, tree, matching and star backbones). We show here that for split graphs and matching or star backbones, $\ell$ is at most a small additive constant (depending on $\gamma$) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on $\ell$ than the previously known bounds.

AB - We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph $G = (V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\gamma$-backbone coloring for $G$ and $H$ is a proper vertex coloring $V\to \{1,2,\ldots\}$ of $G$ in which the colors assigned to adjacent vertices in $H$ differ by at least $\gamma$. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number $\ell$ of colors, for which such colorings $V\to \{1,2,\ldots, \ell\}$ exist, in the worst case is a factor times the chromatic number (for path, tree, matching and star backbones). We show here that for split graphs and matching or star backbones, $\ell$ is at most a small additive constant (depending on $\gamma$) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on $\ell$ than the previously known bounds.

KW - EWI-15830

KW - METIS-263962

KW - IR-67842

U2 - 10.7151/dmgt.1437

DO - 10.7151/dmgt.1437

M3 - Article

VL - 29

SP - 143

EP - 162

JO - Discussiones mathematicae. Graph theory

JF - Discussiones mathematicae. Graph theory

SN - 1234-3099

IS - 1

ER -