### Abstract

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)l ≤ c for all uv ∈ E, is it possible to find a "list coloring," i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ for all uv ∈ E? We prove that every graph of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ √5.437cΔ. Apart from a multiplicative constant, the result is tight, as lists of length √0.5cΔ may be necessary. Moreover, for G = K_{n} (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + 0(1))√n. For planar graphs and c = 1, lists of length 4 suffice.

Original language | English |
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Pages (from-to) | 43-49 |

Number of pages | 7 |

Journal | Journal of Graph Theory |

Volume | 27 |

Issue number | 1 |

Publication status | Published - Jan 1 1998 |

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### Keywords

- Choosability
- Coloring
- Graph
- List coloring

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*27*(1), 43-49.