### Abstract

An L-list colouring of a graph G is a proper vertex colouring in which every vertex v gets a colour from a prescribed list L(v) of allowed colours. Albertson has posed the following problem: Suppose G is a planar graph and each vertex of G has been assigned a list of five colours. Let W C V(G) such that the distance between any two vertices of W is at least d (=4). Can any list colouring of W be extended to a list colouring of G ? We give a construction satisfying the assumptions for d = 4 where the required extension is not possible. As an even stronger property, in our example one can assign lists L(v) to the vertices of G with |L(t')|=3 for v£ W and |I(i>)| = 5 otherwise, such that an Z,-list colouring is not possible. The existence of such graphs is in sharp contrast with Thomassen's theorem stating that a list colouring is always possible if the vertices of 3-element lists belong to the same face of G (and the other lists have 5 colours each).

Original language | English |
---|---|

Pages (from-to) | 169-172 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 251 |

Issue number | 1-3 |

Publication status | Published - May 28 2002 |

### Fingerprint

### Keywords

- Distance
- Graph
- List colouring
- Planar

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*251*(1-3), 169-172.

**A note on planar 5-list colouring : Non-extendability at distance 4.** / Tuza, Z.; Voigt, M.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 251, no. 1-3, pp. 169-172.

}

TY - JOUR

T1 - A note on planar 5-list colouring

T2 - Non-extendability at distance 4

AU - Tuza, Z.

AU - Voigt, M.

PY - 2002/5/28

Y1 - 2002/5/28

N2 - An L-list colouring of a graph G is a proper vertex colouring in which every vertex v gets a colour from a prescribed list L(v) of allowed colours. Albertson has posed the following problem: Suppose G is a planar graph and each vertex of G has been assigned a list of five colours. Let W C V(G) such that the distance between any two vertices of W is at least d (=4). Can any list colouring of W be extended to a list colouring of G ? We give a construction satisfying the assumptions for d = 4 where the required extension is not possible. As an even stronger property, in our example one can assign lists L(v) to the vertices of G with |L(t')|=3 for v£ W and |I(i>)| = 5 otherwise, such that an Z,-list colouring is not possible. The existence of such graphs is in sharp contrast with Thomassen's theorem stating that a list colouring is always possible if the vertices of 3-element lists belong to the same face of G (and the other lists have 5 colours each).

AB - An L-list colouring of a graph G is a proper vertex colouring in which every vertex v gets a colour from a prescribed list L(v) of allowed colours. Albertson has posed the following problem: Suppose G is a planar graph and each vertex of G has been assigned a list of five colours. Let W C V(G) such that the distance between any two vertices of W is at least d (=4). Can any list colouring of W be extended to a list colouring of G ? We give a construction satisfying the assumptions for d = 4 where the required extension is not possible. As an even stronger property, in our example one can assign lists L(v) to the vertices of G with |L(t')|=3 for v£ W and |I(i>)| = 5 otherwise, such that an Z,-list colouring is not possible. The existence of such graphs is in sharp contrast with Thomassen's theorem stating that a list colouring is always possible if the vertices of 3-element lists belong to the same face of G (and the other lists have 5 colours each).

KW - Distance

KW - Graph

KW - List colouring

KW - Planar

UR - http://www.scopus.com/inward/record.url?scp=33750699684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750699684&partnerID=8YFLogxK

M3 - Article

VL - 251

SP - 169

EP - 172

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -