### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- graph coloring
- k -minor-free
- list coloring
- planar

### ASJC Scopus subject areas

- Geometry and Topology

### 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

**List colorings of K _{5}-minor-free graphs with special list assignments.** / Cranston, Daniel W.; Pruchnewski, Anja; Tuza, Z.; Voigt, Margit.

Research output: Contribution to journal › Article

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

*Journal of Graph Theory*, vol. 71, no. 1, pp. 18-30. https://doi.org/10.1002/jgt.20628

_{5}-minor-free graphs with special list assignments. Journal of Graph Theory. 2012 Sep;71(1):18-30. https://doi.org/10.1002/jgt.20628

}

TY - JOUR

T1 - List colorings of K 5-minor-free graphs with special list assignments

AU - Cranston, Daniel W.

AU - Pruchnewski, Anja

AU - Tuza, Z.

AU - Voigt, Margit

PY - 2012/9

Y1 - 2012/9

N2 - 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].

AB - 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].

KW - graph coloring

KW - k -minor-free

KW - list coloring

KW - planar

UR - http://www.scopus.com/inward/record.url?scp=84864462189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864462189&partnerID=8YFLogxK

U2 - 10.1002/jgt.20628

DO - 10.1002/jgt.20628

M3 - Article

VL - 71

SP - 18

EP - 30

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -