### Abstract

The following question was raised by Bruce Richter. Let G be a planar, 3-connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L-list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K_{5}-minor-free r-connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L-list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[S_{k}], where S_{k}: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(S_{k}) in G between the components of G[S _{k}].

Original language | English |
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Pages (from-to) | 18-30 |

Number of pages | 13 |

Journal | Journal of Graph Theory |

Volume | 71 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 2012 |

### Keywords

- graph coloring
- list coloring
- planar

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

_{5}-minor-free graphs with special list assignments.

*Journal of Graph Theory*,

*71*(1), 18-30. https://doi.org/10.1002/jgt.20628