### Abstract

Abstract A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. It is known that planar graphs are (4, 1)-choosable but it is not known if planar graphs are (3, 1)-choosable. We strengthen the result that planar graphs are (4, 1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4. Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors.

Original language | English (US) |
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Article number | 10037 |

Pages (from-to) | 1779-1783 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 338 |

Issue number | 10 |

DOIs | |

State | Published - Sep 30 2015 |

### Keywords

- Choosability with separation
- Graph coloring
- List coloring
- Planar graphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*338*(10), 1779-1783. [10037]. https://doi.org/10.1016/j.disc.2015.01.008